6. Systems of Equations and Inequalities

Solve Linear Systems by Addition

Systems of Linear Equations in Two Variables

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

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;">

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;">

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

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

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

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

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

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

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 systems of equations: using substitution (3)

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

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;">

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?

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 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;">

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

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;">

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;">

Solve a Linear System Graphical Approach

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;">

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 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

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Graph a Linear Inequality in Two Variables

Graph a Nonlinear Inequality in Two Variables

Use Mathematical Models Involving Linear Inequalities

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Systems of Inequalities

Determine Whether an Ordered Pair is a Solution of a Linear System

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Write an Objective Function Describing a Quantity That Must Be Maximized or Minimized

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Solve Nonlinear Systems by Substitution

Solve Nonlinear Systems by Addition

Solve Problems Using Systems of Nonlinear Equations

Systems of Nonlinear Equations in Two Variables

Verify Solution of a System of Linear Equations in 3 Variables

Solve Systems of Linear Equations in Three Variables

Solve Problems Using Systems in 3 Variables

Systems of Linear Equations in 3 Variables

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