1. Fundamental Concepts of Algebra

2. Equations and Inequalities

3. Functions and Graphs

4. Polynomial and Rational Functions

5. Exponential and Logarithmic Functions

6. Systems of Equations and Inequalities

7. Matrices and Determinants

8. Conic Sections

9. Sequences, Induction, and Probability

undefined - Channels

Use the substitution or elimination method to solve each system of equations. Identify any inconsistent systems or systems with infinitely many solutions. If a system has infinitely many solutions, write the solution set with y arbitrary. 
2x + 6y = 6 
5x + 9y = 9

Solve each system by substitution. See Example 1.
7x - y = -10 
3y - x = 10

Solve each system. (Hint: In Exercises 69–72, let 1/x = t and 1/y = u.)
2/x + 1/y = 3/2
3/x - 1/y = 1

Determine the system of equations illustrated in each graph. Write equations in standard form. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/7363c8edfc80a7bd-1685535855985.jpg" alt="" />

In Exercises 1–4, determine whether the given ordered pair is a solution of the system.

(2, 5)
2x + 3y = 17
x + 4y = 16<img class="fr-dib" src="https://lightcat-files.s3.amazonaws.com/problem_images/b2f7e3f00b49fe74-1678180661240.jpg" style="width: 448px;">

In Exercises 1–5, solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets.

In Exercises 31–42, solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. 

x = 9-2y
x + 2y = 13

Solve systems of equations with infinite solutions

Systems of Linear Equations - Inconsistent Systems Using Elimination by Addition - Example 3

In Exercises 31–42, solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. 

2x = 3y + 4
4x = 3 - 5y

In Exercises 5–18, solve each system by the substitution method.

x + 3y = 8
y = 2x - 9<img class="fr-dib" src="https://lightcat-files.s3.amazonaws.com/problem_images/69199754421cab8e-1678181159227.jpg" style="width: 172px;">

Systems of Linear Equations - Inconsistent Systems Using Elimination by Addition - Example 2

Solve each system by elimination. In systems with fractions, first clear denominators. See Example 2.
2x - 3y = -7 
5x + 4y = 17

System of 3 Equations, 3 Unknowns Using Substitution - Basic Example

In Exercises 1–4, determine whether the given ordered pair is a solution of the system.

(- 3, 5)

9x + 7y = 8
8x - 9y = - 69<img class="fr-dib" src="https://lightcat-files.s3.amazonaws.com/problem_images/a04079949943fe37-1678180632076.jpg" style="width: 437px;">

Solve each system. (Hint: In Exercises 69–72, let 1/x = t and 1/y = u.)
2/x + 3/y = 18 
4/x - 5/y = -8

Solving a System of Equations Involving 3 Variables Using Elimination by Addition - Example 3

Solve each system of equations. State whether it is an inconsistent system or has infinitely many solutions. If a system has infinitely many solutions, write the solution set with x arbitrary. See Examples 3 and 4.
5x - 5y - 3 = 0 
x - y - 12 = 0

Solve each system by substitution. See Example 1. 
4x + 3y = -13
-x + y = 5

In Exercises 1–4, determine whether the given ordered pair is a solution of the system.

(2, 3)

x + 3y = 11
x - 5y = - 13<img class="fr-dib" src="https://lightcat-files.s3.amazonaws.com/problem_images/1e7e68538b9c60c2-1678180600617.jpg" style="width: 376px;">

In Exercises 49–50, solve each system for x and y, expressing either value in terms of a or b, if necessary. Assume that a ≠ 0, b ≠ 0 

5ax + 4y = 17
ax + 7y = 22<img class="fr-dib" src="https://lightcat-files.s3.amazonaws.com/problem_images/c50ee8f8bf506449-1678232871383.jpg" style="width: 300px;">

In Exercises 31–42, solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. 

3x - 2y = − 5
4x + y = 8

Use a system of equations to solve each problem. See Example 8. Find an equation of the parabola y = ax^2 + bx + c that passes through the points (2, 3), (-1, 0), and (-2, 2).

Linear System of Equations with Infinitely Many Solutions

In Exercises 19–30, solve each system by the addition method.

4x + 3y = 15
2x - 5y = 1<img class="fr-dib" src="https://lightcat-files.s3.amazonaws.com/problem_images/c2a343c752b7a226-1678232695953.jpg" style="width: 300px;">

Solving Systems of Two Equations and Two Unknowns: Graphing, Substitution, and Elimination

In Exercises 47–48, solve each system by the method of your choice. 

(x + 2)/2 - (y + 4)/3 = 3
(x + y)/5 = (x - y)/2 - 5/2<img class="fr-dib" src="https://lightcat-files.s3.amazonaws.com/problem_images/78838ec51c0402e7-1678232803882.jpg" style="width: 300px;">

Solving Systems of Equations Using Elimination By Addition

Solve each system by substitution. See Example 1.
x - 5y = 8 
x = 6y

Model a real-life situation using a system of linear equations

Systems of Linear Equations - Inconsistent Systems Using Elimination by Addition - Example 1

Solve Linear Systems by Addition

Systems of Linear Equations in Two Variables

Solving a System of Equations Involving 3 Variables Using Elimination by Addition - Example 1

Solve each system by elimination. In systems with fractions, first clear denominators. See Example 2.
(2x-1)/3 + (y+2)/4 = 4
(x+3)/2 - (x-y)/2 = 3

Exercises 86–88 will help you prepare for the material covered in the first section of the next chapter.
a. Does (4, −1) satisfy x + 2y = 2?
b. Does (4, -1) satisfy x- 2y= 6?

Use a system of linear equations to solve Exercises 73–84. How many ounces of a 50% alcohol solution must be mixed with 80 ounces of a 20% alcohol solution to make a 40% alcohol solution?

In Exercises 31–42, solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. 

y = 3x - 5
21x - 35 = 7y

Solve each system by elimination. In systems with fractions, first clear denominators. See Example 2.
x/2+ y/3 = 4
3x/2+3y/2 = 15

Solve systems of equations: using substitution (3)

Homogeneous Systems of Linear Equations - Trivial and Nontrivial Solutions, Part 1

Solve each system by elimination. In systems with fractions, first clear denominators. See Example 2.
6x + 7y + 2 = 0
7x - 6y - 26 = 0

Solve each system by substitution. See Example 1.
3y = 5x + 6 
x + y = 2

In Exercises 5–18, solve each system by the substitution method.

2x - 3y = 8 - 2x
3x + 4y = x + 3y + 14<img class="fr-dib" src="https://lightcat-files.s3.amazonaws.com/problem_images/8b023cdaed243792-1678181390059.jpg" style="width: 300px;">

In Exercises 47–48, solve each system by the method of your choice. 

(x - y)/3 = (x + y)/2 - 1/2
(x + 2)/2 - 4 = (y + 4)/3<img class="fr-dib" src="https://lightcat-files.s3.amazonaws.com/problem_images/eeee483bf3f7b375-1678232836050.jpg" style="width: 300px;">

Determine whether a point is a solution to a linear equation

In Exercises 5–18, solve each system by the substitution method.

y = (1/3)x + 2/3
y = (5/7)x - 2<img class="fr-dib" src="https://lightcat-files.s3.amazonaws.com/problem_images/974e5e2140ca2edf-1678181422678.jpg" style="width: 247px;">

Solve each system by substitution. See Example 1.
-2x = 6y + 18 
-29 = 5y - 3x

For what value(s) of k will the following system of linear equations have no solution? infinitely many solutions? x - 2y = 3
-2x + 4y = k

In Exercises 5–18, solve each system by the substitution method.

x + y = 4
y = 3x<img class="fr-dib" src="https://lightcat-files.s3.amazonaws.com/problem_images/91b9189d817f2fff-1678181105247.jpg" style="width: 178px;">

Use a system of linear equations to solve Exercises 73–84. How many ounces of a 15% alcohol solution must be mixed with 4 ounces of a 20% alcohol solution to make a 17% alcohol solution?

Use the substitution or elimination method to solve each system of equations. Identify any inconsistent systems or systems with infinitely many solutions. If a system has infinitely many solutions, write the solution set with y arbitrary. 
6x + 10y = -11 
9x + 6y = -3

Use a system of equations to solve each problem. See Example 8. Find an equation of the line y = ax + b that passes through the points (-2, 1) and (-1, -2).

A chemist needs to mix a solution that is 34% silver nitrate with one that is 4% silver nitrate to obtain 100 milliliters of a mixture that is 7% silver nitrate. How many milliliters of each of the solutions must be used?

Solve each problem using a system of equations. A company sells recordable CDs for $0.80 each and play-only CDs for $0.60 each. The company receives $76.00 for an order of 100 CDs. However, the customer neglected to specify how many of each type to send. Determine the number of each type of CD that should be sent.

Solve a system of linear equations using elimination

Represent constraints on solutions using systems of linear equations

❖ Solving Linear Systems of Equations Using Substitution ❖

Solving a System of Equations Involving 3 Variables Using Elimination by Addition - Example 2

In Exercises 5–18, solve each system by the substitution method.

x = 4y - 2
x = 6y + 8<img class="fr-dib" src="https://lightcat-files.s3.amazonaws.com/problem_images/c1b68be5ec90eca5-1678181283471.jpg" style="width: 188px;">

Solve each system by elimination. In systems with fractions, first clear denominators. See Example 2.
4x + y = -23
x - 2y = -17

In Exercises 31–42, solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. 

x/4 - y/4 = −1
x + 4y = -9

In Exercises 19–30, solve each system by the addition method.

3x = 4y + 1
3y = 1 - 4x<img class="fr-dib" src="https://lightcat-files.s3.amazonaws.com/problem_images/82e6429e6c536b78-1678232754875.jpg" style="width: 300px;">

In Exercises 31–42, solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. 

x + 3y = 2
3x + 9y = 6

In Exercises 5–18, solve each system by the substitution method.

2x + 5y = - 4
3x - y = 11<img class="fr-dib" src="https://lightcat-files.s3.amazonaws.com/problem_images/5872268f6eedfde8-1678181353869.jpg" style="width: 249px;">

Solving a Dependent System of Linear Equations involving 3 Variables

Solve each system by elimination. In systems with fractions, first clear denominators. See Example 2.
5x + 7y = 6 
10x - 3y = 46

Solve each system by substitution. See Example 1.
8x - 10y = -22 
3x + y = 6

The perimeter of a table tennis top is 28 feet. The difference between 4 times the length and 3 times the width is 21 feet. Find the dimensions. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/b146006c024eb4bf-1673100248578.jpg" alt="" />

In Exercises 19–30, solve each system by the addition method.

x + y = 1
x - y = 3<img class="fr-dib" src="https://lightcat-files.s3.amazonaws.com/problem_images/0717855bf2c8f48f-1678181489183.jpg" style="width: 193px;">

In Exercises 19–30, solve each system by the addition method.

3x - 4y = 11
2x + 3y = - 4<img class="fr-dib" src="https://lightcat-files.s3.amazonaws.com/problem_images/a506f09a24fbb2c2-1678232725407.jpg" style="width: 300px;">

Determine the system of equations illustrated in each graph. Write equations in standard form. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/78b038018bacbf8a-1683226367471.jpg" alt="" />

In Exercises 19–30, solve each system by the addition method.

2x + 3y = 6
2x - 3y = 6<img class="fr-dib" src="https://lightcat-files.s3.amazonaws.com/problem_images/21db37da8ba5b698-1678232319164.jpg" style="width: 259px;">

In Exercises 43–46, let x represent one number and let y represent the other number. Use the given conditions to write a system of equations. Solve the system and find the numbers. The sum of two numbers is 7. If one number is subtracted from the other, their difference is -1. Find the numbers.

Solve equations with infinite solutions

Solve systems of linear equations by the addition/elimination method

In Exercises 19–30, solve each system by the addition method.

x + 2y = 2
- 4x + 3y = 25<img class="fr-dib" src="https://lightcat-files.s3.amazonaws.com/problem_images/4f60c3bba7a43229-1678232570108.jpg" style="width: 300px;">

Use the substitution or elimination method to solve each system of equations. Identify any inconsistent systems or systems with infinitely many solutions. If a system has infinitely many solutions, write the solution set with y arbitrary. 
1/6x + 1/3y = 8
1/4x + 1/2y = 12

Solve a Linear System Graphical Approach

Solve each system of equations. State whether it is an inconsistent system or has infinitely many solutions. If a system has infinitely many solutions, write the solution set with x arbitrary. See Examples 3 and 4.
9x - 5y = 1 
-18x + 10y = 1

In Exercises 5–18, solve each system by the substitution method.

5x + 2y = 0
x - 3y = 0<img class="fr-dib" src="https://lightcat-files.s3.amazonaws.com/problem_images/3b035531ff8e2035-1678181318467.jpg" style="width: 232px;">

Solve each problem. See Examples 5 and 9. The sum of two numbers is 47, and the difference between the numbers is 1. Find the numbers.

Solving word problems by using systems of equations

Solve systems of equations that have no solution

Solve a Linear System û by the Elimination Method

Solve a Linear System û by the Substitution Method

Solve each problem. See Examples 5 and 9. At the Berger ranch, 6 goats and 5 sheep sell for $305, while 2 goats and 9 sheep sell for $285. Find the cost of a single goat and of a single sheep.

Solve systems of equations: using substitution

Learn the toughest concepts covered in Algebra with step-by-step video tutorials and practice problems by world-class tutors