Chapter 9: Systems of Equations and InequalitiesContent check — Precalculus, Chapter 9: Systems of Equations and Inequalities

Determine whether the ordered pair $\left(8,5\right)$ is a solution to the following system.

Solve the following system of equations by graphing.

Solve the following system of equations by substitution.

Solve the system of equations by addition.

Solve the system of equations by addition.

Solve the following system of equations in two variables.

Solve the following system of equations in two variables.

For the following exercises, determine whether the given ordered pair is a solution to the system of equations.

$$\begin{array}{l}\mathrm{-3}x-5y=13\hfill \\ -x+4y=10\hfill \end{array}$$ and $(\mathrm{-6},1)$

For the following exercises, determine whether the given ordered pair is a solution to the system of equations.

$$\begin{array}{l}\mathrm{-2}x+5y=7\hfill \\ 2x+9y=7\hfill \end{array}$$ and $(\mathrm{-1},1)$

For the following exercises, solve each system by substitution.

$$\begin{array}{l}x+3y=5\hfill \\ 2x+3y=4\hfill \end{array}$$

For the following exercises, solve each system by substitution.

$$\begin{array}{l}4x+2y=\mathrm{-10}\\ 3x+9y=0\end{array}$$

For the following exercises, solve each system by substitution.

$$\begin{array}{l}-2x+3y=1.2\hfill \\ -3x-6y=1.8\hfill \end{array}$$

For the following exercises, solve each system by substitution.

$$\begin{array}{l}3x+5y=9\hfill \\ 30x+50y=\mathrm{-90}\hfill \end{array}$$

For the following exercises, solve each system by substitution.

$$\begin{array}{l}\frac{1}{2}x+\frac{1}{3}y=16\\ \frac{1}{6}x+\frac{1}{4}y=9\end{array}$$

For the following exercises, solve each system by addition.

$$\begin{array}{l}\mathrm{-2}x+5y=\mathrm{-42}\hfill \\ 7x+2y=30\hfill \end{array}$$

For the following exercises, solve each system by addition.

$$\begin{array}{l}5x-y=\mathrm{-2.6}\hfill \\ \mathrm{-4}x\mathrm{-6}y=1.4\hfill \end{array}$$

For the following exercises, solve each system by addition.

$$\begin{array}{l}\mathrm{-x}+2y=\mathrm{-1}\hfill \\ 5x\mathrm{-10}y=6\hfill \end{array}$$

For the following exercises, solve each system by addition.

$$\begin{array}{l}\frac{5}{6}x+\frac{1}{4}y=0\\ \frac{1}{8}x-\frac{1}{2}y=-\frac{43}{120}\end{array}$$

For the following exercises, solve each system by addition.

$$\begin{array}{l}\hfill \\ \begin{array}{l}\mathrm{-0.2}x+0.4y=0.6\hfill \\ x\mathrm{-2}y=\mathrm{-3}\hfill \end{array}\hfill \end{array}$$

For the following exercises, solve each system by any method.

$$\begin{array}{l}5x+9y=16\hfill \\ x+2y=4\hfill \end{array}$$

For the following exercises, solve each system by any method.

$$\begin{array}{l}5x\mathrm{-2}y=2.25\\ 7x\mathrm{-4}y=3\end{array}$$

For the following exercises, solve each system by any method.

$$\begin{array}{l}7x\mathrm{-4}y=\frac{7}{6}\hfill \\ 2x+4y=\frac{1}{3}\hfill \end{array}$$

For the following exercises, solve each system by any method.

$$\begin{array}{l}\frac{7}{3}x-\frac{1}{6}y=2\hfill \\ -\frac{21}{6}x+\frac{3}{12}y=\mathrm{-3}\hfill \end{array}$$

For the following exercises, solve each system by any method.

$$\begin{array}{l}2.2x+1.3y=\mathrm{-0.1}\\ 4.2x+4.2y=2.1\end{array}$$

For the following exercises, graph the system of equations and state whether the system is consistent, inconsistent, or dependent and whether the system has one solution, no solution, or infinite solutions.

$$\begin{array}{l}3x-y=0.6\\ x\mathrm{-2}y=1.3\end{array}$$

For the following exercises, graph the system of equations and state whether the system is consistent, inconsistent, or dependent and whether the system has one solution, no solution, or infinite solutions.

$$\begin{array}{l}x+2y=7\hfill \\ 2x+6y=12\hfill \end{array}$$

For the following exercises, graph the system of equations and state whether the system is consistent, inconsistent, or dependent and whether the system has one solution, no solution, or infinite solutions.

$$\begin{array}{l}3x\mathrm{-2}y=5\hfill \\ \mathrm{-9}x+6y=\mathrm{-15}\hfill \end{array}$$

For the following exercises, use the intersect function on a graphing device to solve each system. Round all answers to the nearest hundredth.

$$\begin{array}{l}\mathrm{-0.01}x+0.12y=0.62\hfill \\ 0.15x+0.20y=0.52\hfill \end{array}$$

For the following exercises, use the intersect function on a graphing device to solve each system. Round all answers to the nearest hundredth.

$$\begin{array}{l}0.15x+0.27y=0.39\hfill \\ \mathrm{-0.34}x+0.56y=1.8\hfill \end{array}$$

For the following exercises, solve each system in terms of $A,B,C,D,E,$ and $F$ where $A–F$ are nonzero numbers. Note that $A\ne B$ and $AE\ne BD.$

$$\begin{array}{l}x+y=A\\ x-y=B\end{array}$$

For the following exercises, solve each system in terms of $A,B,C,D,E,$ and $F$ where $A–F$ are nonzero numbers. Note that $A\ne B$ and $AE\ne BD.$

$$\begin{array}{l}Ax+y=0\\ Bx+y=1\end{array}$$

For the following exercises, solve each system in terms of $A,B,C,D,E,$ and $F$ where $A–F$ are nonzero numbers. Note that $A\ne B$ and $AE\ne BD.$

$$\begin{array}{l}Ax+By=C\\ Dx+Ey=F\end{array}$$

For the following exercises, solve for the desired quantity.

An Ethiopian restaurant has a cost of production $C(x)=$ $11x+$ $120$ and a revenue function $R(x)=5x.$ When does the company start to turn a profit?

For the following exercises, solve for the desired quantity.

A musician charges $C(x)=$ $64x+$ $\mathrm{20,000}$ where $x$ is the total number of attendees at the concert. The venue charges $80 per ticket. After how many people buy tickets does the venue break even, and what is the value of the total tickets sold at that point?

For the following exercises, use a system of linear equations with two variables and two equations to solve.

Find two numbers whose sum is 28 and difference is 13.

For the following exercises, use a system of linear equations with two variables and two equations to solve.

The startup cost for a restaurant is $120,000, and each meal costs $10 for the restaurant to make. If each meal is then sold for $15, after how many meals does the restaurant break even?

For the following exercises, use a system of linear equations with two variables and two equations to solve.

A total of 1,595 first- and second-year college students gathered at a pep rally. The number of first-years exceeded the number of second-years by 15. How many students from each year group were in attendance?

For the following exercises, use a system of linear equations with two variables and two equations to solve.

There were 130 faculty at a conference. If there were 18 more women than men attending, how many of each gender attended the conference?

For the following exercises, use a system of linear equations with two variables and two equations to solve.

If a scientist mixed 10% saline solution with 60% saline solution to get 25 gallons of 40% saline solution, how many gallons of 10% and 60% solutions were mixed?

For the following exercises, use a system of linear equations with two variables and two equations to solve.

An investor invested 1.1 million dollars into two land investments. On the first investment, Swan Peak, her return was a 110% increase on the money she invested. On the second investment, Riverside Community, she earned 50% over what she invested. If she earned $1 million in profits, how much did she invest in each of the land deals?

For the following exercises, use a system of linear equations with two variables and two equations to solve.

If an investor invests $23,000 into two bonds, one that pays 4% in simple interest, and the other paying 2% simple interest, and the investor earns $710.00 annual interest, how much was invested in each account?

For the following exercises, use a system of linear equations with two variables and two equations to solve.

A store clerk sold 60 pairs of sneakers. The high-tops sold for $98.99 and the low-tops sold for $129.99. If the receipts for the two types of sales totaled $6,404.40, how many of each type of sneaker were sold?

For the following exercises, use a system of linear equations with two variables and two equations to solve.

Admission into an amusement park for 4 children and 2 adults is $116.90. For 6 children and 3 adults, the admission is $175.35. Assuming a different price for children and adults, what is the price of the child’s ticket and the price of the adult ticket?

Solve the system of equations in three variables.

Solve the system of three equations in three variables.

Solve the following system.

For the following exercises, determine whether the ordered triple given is the solution to the system of equations.

$$\begin{array}{l}6x-y+3z=6\hfill \\ 3x+5y+2z=0\hfill \\ x+y=0\hfill \end{array}$$ and $(3,\mathrm{-3},\mathrm{-5})$

For the following exercises, determine whether the ordered triple given is the solution to the system of equations.

$$\begin{array}{l}x-y=0\hfill \\ x-z=5\hfill \\ x-y+z=\mathrm{-1}\hfill \end{array}$$ and $(4,4,\mathrm{-1})$

For the following exercises, solve each system by elimination.

$$\begin{array}{l}3x\mathrm{-4}y+2z=\mathrm{-15}\hfill \\ 2x+4y+z=16\hfill \\ 2x+3y+5z=20\hfill \end{array}$$

For the following exercises, solve each system by elimination.

$$\begin{array}{l}5x+2y+4z=9\hfill \\ \mathrm{-3}x+2y+z=10\hfill \\ 4x\mathrm{-3}y+5z=\mathrm{-3}\hfill \end{array}$$

For the following exercises, solve each system by elimination.

$$\begin{array}{l}5x\mathrm{-2}y+3z=4\hfill \\ \mathrm{-4}x+6y\mathrm{-7}z=\mathrm{-1}\hfill \\ 3x+2y-z=4\hfill \end{array}$$

For the following exercises, solve each system by Gaussian elimination.

$$\begin{array}{l}2x-y+3z=17\hfill \\ \mathrm{-5}x+4y\mathrm{-2}z=\mathrm{-46}\hfill \\ 2y+5z=\mathrm{-7}\hfill \end{array}$$

For the following exercises, solve each system by Gaussian elimination.

$$\begin{array}{l}2x+3y\mathrm{-6}z=1\hfill \\ \mathrm{-4}x\mathrm{-6}y+12z=\mathrm{-2}\hfill \\ x+2y+5z=10\hfill \end{array}$$

For the following exercises, solve each system by Gaussian elimination.

$$\begin{array}{l}2x+3y\mathrm{-4}z=5\hfill \\ \mathrm{-3}x+2y+z=11\hfill \\ -x+5y+3z=4\hfill \end{array}$$

For the following exercises, solve each system by Gaussian elimination.

$$\begin{array}{l}x+y+z=14\hfill \\ 2y+3z=\mathrm{-14}\hfill \\ \mathrm{-16}y\mathrm{-24}z=\mathrm{-112}\hfill \end{array}$$

For the following exercises, solve each system by Gaussian elimination.

$$\begin{array}{l}x+y+z=0\hfill \\ 2x-y+3z=0\hfill \\ x-z=0\hfill \end{array}$$

For the following exercises, solve each system by Gaussian elimination.

$$\begin{array}{l}x+y+z=0\hfill \\ 2x-y+3z=0\hfill \\ x-z=1\hfill \end{array}$$

For the following exercises, solve each system by Gaussian elimination.

$$\begin{array}{l}6x\mathrm{-5}y+6z=38\hfill \\ \frac{1}{5}x-\frac{1}{2}y+\frac{3}{5}z=1\hfill \\ \mathrm{-4}x-\frac{3}{2}y-z=\mathrm{-74}\hfill \end{array}$$

For the following exercises, solve each system by Gaussian elimination.

$$\begin{array}{l}-\frac{1}{3}x-\frac{1}{2}y-\frac{1}{4}z=\frac{3}{4}\\ -\frac{1}{2}x-\frac{1}{4}y-\frac{1}{2}z=2\hfill \\ -\frac{1}{4}x-\frac{3}{4}y-\frac{1}{2}z=-\frac{1}{2}\hfill \end{array}$$

For the following exercises, solve each system by Gaussian elimination.

$$\begin{array}{l}\frac{4}{5}x-\frac{7}{8}y+\frac{1}{2}z=1\hfill \\ -\frac{4}{5}x-\frac{3}{4}y+\frac{1}{3}z=\mathrm{-8}\hfill \\ -\frac{2}{5}x-\frac{7}{8}y+\frac{1}{2}z=\mathrm{-5}\hfill \end{array}$$

For the following exercises, solve each system by Gaussian elimination.

$$\begin{array}{l}\\ -\frac{1}{4}x-\frac{5}{4}y+\frac{5}{2}z=\mathrm{-5}\\ -\frac{1}{2}x-\frac{5}{3}y+\frac{5}{4}z=\frac{55}{12}\hfill \\ -\frac{1}{3}x-\frac{1}{3}y+\frac{1}{3}z=\frac{5}{3}\hfill \end{array}$$

For the following exercises, solve each system by Gaussian elimination.

$$\begin{array}{l}0.1x\mathrm{-0.2}y+0.3z=2\\ 0.5x\mathrm{-0.1}y+0.4z=8\\ 0.7x\mathrm{-0.2}y+0.3z=8\end{array}$$

For the following exercises, solve each system by Gaussian elimination.

$$\begin{array}{l}1.1x+0.7y\mathrm{-3.1}z=\mathrm{-1.79}\\ 2.1x+0.5y\mathrm{-1.6}z=\mathrm{-0.13}\\ 0.5x+0.4y\mathrm{-0.5}z=\mathrm{-0.07}\end{array}$$

For the following exercises, solve each system by Gaussian elimination.

$$\begin{array}{l}0.1x+0.2y+0.3z=0.37\\ 0.1x\mathrm{-0.2}y\mathrm{-0.3}z=\mathrm{-0.27}\\ 0.5x\mathrm{-0.1}y\mathrm{-0.3}z=\mathrm{-0.03}\end{array}$$

For the following exercises, solve each system by Gaussian elimination.

$$\begin{array}{l}0.5x+0.2y\mathrm{-0.3}z=1\\ 0.4x\mathrm{-0.6}y+0.7z=0.8\\ 0.3x\mathrm{-0.1}y\mathrm{-0.9}z=0.6\end{array}$$

For the following exercises, solve each system by Gaussian elimination.

$$\begin{array}{l}0.8x+0.8y+0.8z=2.4\\ 0.3x\mathrm{-0.5}y+0.2z=0\\ 0.1x+0.2y+0.3z=0.6\end{array}$$

For the following exercises, solve the system for $x,y,$ and $z.$

$$\begin{array}{l}5x\mathrm{-3}y-\frac{z+1}{2}=\frac{1}{2}\hfill \\ 6x+\frac{y\mathrm{-9}}{2}+2z=\mathrm{-3}\hfill \\ \frac{x+8}{2}\mathrm{-4}y+z=4\hfill \end{array}$$

For the following exercises, solve the system for $x,y,$ and $z.$

$$\begin{array}{l}\frac{x\mathrm{-3}}{6}+\frac{y+2}{2}-\frac{z\mathrm{-3}}{3}=2\\ \frac{x+2}{4}+\frac{y\mathrm{-5}}{2}+\frac{z+4}{2}=1\\ \frac{x+6}{2}-\frac{y\mathrm{-3}}{2}+z+1=9\end{array}$$

For the following exercises, solve the system for $x,y,$ and $z.$

Three even numbers sum up to 108. The smaller is half the larger and the middle number is $\frac{3}{4}$ the larger. What are the three numbers?

For the following exercises, solve the system for $x,y,$ and $z.$

At a family reunion, there were only blood relatives, consisting of children, parents, and grandparents, in attendance. There were 400 people total. There were twice as many parents as grandparents, and 50 more children than parents. How many children, parents, and grandparents were in attendance?

For the following exercises, solve the system for $x,y,$ and $z.$

Your roommate, Shani, offered to buy groceries for you and your other roommate. The total bill was $82. She forgot to save the individual receipts but remembered that your groceries were $0.05 cheaper than half of her groceries, and that your other roommate’s groceries were $2.10 more than your groceries. How much was each of your share of the groceries?

For the following exercises, solve the system for $x,y,$ and $z.$

Three coworkers work for the same employer. Their jobs are warehouse manager, office manager, and truck driver. The sum of the annual salaries of the warehouse manager and office manager is $82,000. The office manager makes $4,000 more than the truck driver annually. The annual salaries of the warehouse manager and the truck driver total $78,000. What is the annual salary of each of the co-workers?

For the following exercises, solve the system for $x,y,$ and $z.$

A local band sells out for their concert. They sell all 1,175 tickets for a total purse of $28,112.50. The tickets were priced at $20 for student tickets, $22.50 for children, and $29 for adult tickets. If the band sold twice as many adult as children tickets, how many of each type was sold?

For the following exercises, solve the system for $x,y,$ and $z.$

Last year, at Haven’s Pond Car Dealership, for a particular model of BMW, Jeep, and Toyota, one could purchase all three cars for a total of $140,000. This year, due to inflation, the same cars would cost $151,830. The cost of the BMW increased by 8%, the Jeep by 5%, and the Toyota by 12%. If the price of last year’s Jeep was $7,000 less than the price of last year’s BMW, what was the price of each of the three cars last year?

For the following exercises, solve the system for $x,y,$ and $z.$

You inherit one million dollars. You invest it all in three accounts for one year. The first account pays 3% compounded annually, the second account pays 4% compounded annually, and the third account pays 2% compounded annually. After one year, you earn $34,000 in interest. If you invest four times the money into the account that pays 3% compared to 2%, how much did you invest in each account?

For the following exercises, solve the system for $x,y,$ and $z.$

The top three countries in oil consumption in a certain year are as follows: the United States, Japan, and China. In millions of barrels per day, the three top countries consumed 39.8% of the world’s consumed oil. The United States consumed 0.7% more than four times China’s consumption. The United States consumed 5% more than triple Japan’s consumption. What percent of the world oil consumption did the United States, Japan, and China consume?¹

For the following exercises, solve the system for $x,y,$ and $z.$

The top three sources of oil imports for the United States in the same year were Saudi Arabia, Mexico, and Canada. The three top countries accounted for 47% of oil imports. The United States imported 1.8% more from Saudi Arabia than they did from Mexico, and 1.7% more from Saudi Arabia than they did from Canada. What percent of the United States oil imports were from these three countries?³

For the following exercises, solve the system for $x,y,$ and $z.$

At one time, in the United States, 398 species of animals were on the endangered species list. The top groups were mammals, birds, and fish, which comprised 55% of the endangered species. Birds accounted for 0.7% more than fish, and fish accounted for 1.5% more than mammals. What percent of the endangered species came from mammals, birds, and fish?

Solve the given system of equations by substitution.

Solve the system of nonlinear equations.

Find the solution set for the given system of nonlinear equations.

Graph the given system of inequalities.

For the following exercises, solve the system of nonlinear equations using substitution.

$$\begin{array}{l}y=x\mathrm{-3}\hfill \\ x^2+y^2=9\hfill \end{array}$$

For the following exercises, solve the system of nonlinear equations using substitution.

$$\begin{array}{l}y=-x\hfill \\ x^2+y^2=9\hfill \end{array}$$

For the following exercises, solve the system of nonlinear equations using elimination.

$$\begin{array}{l}4x^2\mathrm{-9}{y}^2=36\hfill \\ 4x^2+9y^2=36\hfill \end{array}$$

For the following exercises, solve the system of nonlinear equations using elimination.

$$\begin{array}{l}2x^2+4y^2=4\hfill \\ 2x^2\mathrm{-4}{y}^2=25x\mathrm{-10}\hfill \end{array}$$

For the following exercises, solve the system of nonlinear equations using elimination.

$$\begin{array}{l}{x}^2+y^2+\frac{1}{16}=2500\\ y=2x^2\end{array}$$

For the following exercises, use any method to solve the system of nonlinear equations.

$$\begin{array}{l}-x^2+y=2\\ -x+y=2\hfill \end{array}$$

For the following exercises, use any method to solve the system of nonlinear equations.

$$\begin{array}{l}{x}^2+y^2=1\hfill \\ y=-x^2\hfill \end{array}$$

For the following exercises, use any method to solve the system of nonlinear equations.

$$\begin{array}{l}9x^2+25y^2=225\hfill \\ {(x\mathrm{-6})}^2+y^2=1\hfill \end{array}$$

For the following exercises, use any method to solve the system of nonlinear equations.

$$\begin{array}{l}2x^3-x^2=y\hfill \\ x^2+y=0\hfill \end{array}$$

For the following exercises, use any method to solve the nonlinear system.

$$\begin{array}{l}{x}^2-y^2=9\hfill \\ x=3\hfill \end{array}$$

For the following exercises, use any method to solve the nonlinear system.

$$\begin{array}{l}{x}^2-y^2=9\hfill \\ x-y=0\hfill \end{array}$$

For the following exercises, use any method to solve the nonlinear system.

$$\begin{array}{l}-x^2+y=2\hfill \\ 2y=-x\hfill \end{array}$$

For the following exercises, use any method to solve the nonlinear system.

$$\begin{array}{l}{x}^2+y^2=1\hfill \\ y^2=x^2\hfill \end{array}$$

For the following exercises, use any method to solve the nonlinear system.

$$\begin{array}{l}3x^2-y^2=12\hfill \\ {(x\mathrm{-1})}^2+y^2=1\hfill \end{array}$$

For the following exercises, use any method to solve the nonlinear system.

$$\begin{array}{l}3x^2-y^2=12\hfill \\ x^2+y^2=16\hfill \end{array}$$

For the following exercises, use any method to solve the nonlinear system.

$$\begin{array}{l}{x}^2+y^2\mathrm{-6}y=7\hfill \\ x^2+y=1\hfill \end{array}$$

For the following exercises, graph the inequality.

$x^2+y<9$

For the following exercises, graph the system of inequalities. Label all points of intersection.

$$\begin{array}{l}{x}^2+y<1\\ y>2x\end{array}$$

For the following exercises, graph the system of inequalities. Label all points of intersection.

$$\begin{array}{l}{x}^2+y^2<25\\ 3x^2-y^2>12\end{array}$$

For the following exercises, graph the system of inequalities. Label all points of intersection.

$$\begin{array}{l}{x}^2+3y^2>16\\ 3x^2-y^2<1\end{array}$$

For the following exercises, graph the inequality.

$$\begin{array}{l}y\le -\log (x)\\ y\le e^x\end{array}$$

For the following exercises, find the solutions to the nonlinear equations with two variables.

$$\begin{array}{c}\frac{6}{x^2}-\frac{1}{y^2}=8\\ \frac{1}{x^2}-\frac{6}{y^2}=\frac{1}{8}\end{array}$$

For the following exercises, find the solutions to the nonlinear equations with two variables.

$$\begin{array}{l}{x}^2-xy\mathrm{-2}{y}^2\mathrm{-6}=0\hfill \\ x^2+y^2=1\hfill \end{array}$$

For the following exercises, solve the system of inequalities. Use a calculator to graph the system to confirm the answer.

$$\begin{array}{l}xy<1\\ y>\sqrt{x}\end{array}$$

For the following exercises, construct a system of nonlinear equations to describe the given behavior, then solve for the requested solutions.

Two numbers add up to 300. One number is twice the square of the other number. What are the numbers?

For the following exercises, construct a system of nonlinear equations to describe the given behavior, then solve for the requested solutions.

A laptop company has discovered their cost and revenue functions for each day: $C(x)=$ $3x^2\mathrm{-10}x+$ $200$ and $R(x)=$ $\mathrm{-2}{x}^2+$ $100x+$ $50.$ If they want to make a profit, what is the range of laptops per day that they should produce? Round to the nearest number which would generate profit.

Find the partial fraction decomposition of the following expression.

Find the partial fraction decomposition of the expression with repeated linear factors.

Find the partial fraction decomposition of the expression with a nonrepeating irreducible quadratic factor.

Find the partial fraction decomposition of the expression with a repeated irreducible quadratic factor.

For the following exercises, find the decomposition of the partial fraction for the nonrepeating linear factors.

$\frac{3x\mathrm{-79}}{x^2\mathrm{-5}x\mathrm{-24}}$

For the following exercises, find the decomposition of the partial fraction for the nonrepeating linear factors.

$\frac{10x+47}{x^2+7x+10}$

For the following exercises, find the decomposition of the partial fraction for the nonrepeating linear factors.

$\frac{32x\mathrm{-11}}{20x^2\mathrm{-13}x+2}$

For the following exercises, find the decomposition of the partial fraction for the nonrepeating linear factors.

$\frac{5x}{x^2\mathrm{-9}}$

For the following exercises, find the decomposition of the partial fraction for the nonrepeating linear factors.

$\frac{6x}{x^2\mathrm{-4}}$

For the following exercises, find the decomposition of the partial fraction for the nonrepeating linear factors.

$\frac{4x\mathrm{-1}}{x^2-x\mathrm{-6}}$

For the following exercises, find the decomposition of the partial fraction for the nonrepeating linear factors.

$\frac{3x\mathrm{-1}}{x^2\mathrm{-5}x+6}$

For the following exercises, find the decomposition of the partial fraction for the repeating linear factors.

$\frac{x}{{\left(x\mathrm{-2}\right)}^2}$

For the following exercises, find the decomposition of the partial fraction for the repeating linear factors.

$\frac{\mathrm{-24}x\mathrm{-27}}{{\left(4x+5\right)}^2}$

For the following exercises, find the decomposition of the partial fraction for the repeating linear factors.

$\frac{5-x}{{\left(x\mathrm{-7}\right)}^2}$

For the following exercises, find the decomposition of the partial fraction for the repeating linear factors.

$\frac{5x^2+20x+8}{2x{\left(x+1\right)}^2}$

For the following exercises, find the decomposition of the partial fraction for the repeating linear factors.

$\frac{54x^3+127x^2+80x+16}{2x^2{\left(3x+2\right)}^2}$

For the following exercises, find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor.

$\frac{4x^2+6x+11}{\left(x+2\right)\left(x^2+x+3\right)}$

For the following exercises, find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor.

$\frac{\mathrm{-2}{x}^2+10x+4}{\left(x\mathrm{-1}\right)\left(x^2+3x+8\right)}$

For the following exercises, find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor.

$\frac{4x^2+17x\mathrm{-1}}{\left(x+3\right)\left(x^2+6x+1\right)}$

For the following exercises, find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor.

$\frac{4x^2+5x+3}{x^3\mathrm{-1}}$

For the following exercises, find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor.

$\frac{3x^2\mathrm{-7}x+33}{x^3+27}$

For the following exercises, find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor.

$\frac{4x^2+4x+12}{8x^3\mathrm{-27}}$

For the following exercises, find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor.

$\frac{\mathrm{-2}{x}^3\mathrm{-30}{x}^2+36x+216}{x^4+216x}$