#### Need Help?

Get in touch with us

# Equations by Graphing

Sep 15, 2022

## Introduction

### System of Linear Equations

A system of linear equations consists of two or more linear equations.

A solution of a system of linear equations in two variables is an ordered pair of numbers that is a solution of both equations in the system.

Example 1:

Determine whether (–3, 1) is a solution of the system.

x – y = – 4

2x + 10y = 4

Solution:

Replace x with –3 and y with 1 in both equations.

First equation: –3 – 1 = – 4 (True)

Second equation: 2(–3) + 10(1) = – 6 + 10 = 4 (True)

Since the point (–3, 1) produces a true statement in both equations, it is a solution of the system.

Since a solution of a system of equations is a solution common to both equations, it is also a point common to the graphs of both equations.

To find the solution of a system of two linear equations, we graph the equations and see where the lines intersect.

### Solve a system of equations by graphing

Example 2:

Solve the system by graphing.

2x – y = 6 (Equation 1)

x + 3y = 10 (Equation 2)

Solution:

First, graph 2xy = 6.

Second, graph x + 3y = 10.

The lines APPEAR to intersect at (4, 2).

Although the solution to the system of equations appears to be (4, 2), you still need to check the answer by substituting x = 4 and y = 2 into the two equations.

First equation:

2(4) – 2 = 8 – 2 = 6      (True)

Second equation:

4 + 3(2) = 4 + 6 = 10    (True)

The point (4, 2) checks, so it is the solution of the system.

### Graph systems of equations with infinitely many solutions or no solution

Example 3:

What is the solution of each system of equations? Use a graph to explain your answer.

–x + 3y = 6

3x – 9y = 9

Solution:

First, graph –x + 3y = 6.

Second, graph 3x – 9y = 9.

The lines APPEAR to be parallel.

Although the lines appear to be parallel, we need to check their slopes.

–x + 3y = 6      First equation

3y = x + 6                 Add x to both sides.

y = 1/3 x + 2              Divide both sides by 3.

3x – 9y = 9    Second equation

–9y = –3x + 9            Subtract 3x from both sides.

y =    1/3 x – 1             Divide both sides by –9.

Both lines have a slope of 1/3, so they are parallel and do not intersect. Hence, there is no solution to the system.

Example 4:

What is the solution of each system of equations? Use a graph to explain your answer.

x = 3y – 1

2x – 6y = –2

Solution:

First, graph x = 3y – 1.

Second, graph 2x – 6y = –2.

The lines APPEAR to be identical.

Although the lines appear to be identical, we need to check that their slopes and y-intercepts are the same.

x = 3y – 1 First equation

3y = x + 1         Add 1 to both sides.

y = 1/3 x + 1/3         Divide both sides by 3.

2x – 6y = – 2            Second equation

–6y = – 2x – 2       Subtract 2x from both sides.

y = 1/3 x + 1/3        Divide both sides by -6.

Any ordered pair that is a solution of one equation is a solution of the other. This means that the system has an infinite number of solutions.

## Exercise

1. A ______________________ consists of two or more linear equations.
2. Determine whether (4, 2) is a solution of the system.
3. Solve the system by graphing

y = x – 1 (Equation 1)

y = −x + 3 (Equation 2)

• Use a graph to solve the following system of equation.

3x + 2y = 9

2/3y = 3 – x

• Use a graph to solve the following system of equation.

y = 1/2x + 7

4x – 8y = 12

• Use a graph to solve the following system of equation.

y = x

y = 2x + 1

• Determine whether the system of equations shown in the graph has no solution or infinitely many solutions.
• Determine whether the system of equations shown in the graph has no solution or infinitely many solutions.

### What have we learned

• Solving a system of linear equations by graphing.
• Graph systems of equations with infinitely many solutions or no solution.

#### Addition and Multiplication Using Counters & Bar-Diagrams

Introduction: We can find the solution to the word problem by solving it. Here, in this topic, we can use 3 methods to find the solution. 1. Add using counters 2. Use factors to get the product 3. Write equations to find the unknown. Addition Equation: 8+8+8 =? Multiplication equation: 3×8=? Example 1: Andrew has […]

#### Dilation: Definitions, Characteristics, and Similarities

Understanding Dilation A dilation is a transformation that produces an image that is of the same shape and different sizes. Dilation that creates a larger image is called enlargement. Describing Dilation Dilation of Scale Factor 2 The following figure undergoes a dilation with a scale factor of 2 giving an image A’ (2, 4), B’ […]

#### How to Write and Interpret Numerical Expressions?

Write numerical expressions What is the Meaning of Numerical Expression? A numerical expression is a combination of numbers and integers using basic operations such as addition, subtraction, multiplication, or division. The word PEMDAS stands for: P → Parentheses E → Exponents M → Multiplication D → Division  A → Addition S → Subtraction         Some examples […]

#### System of Linear Inequalities and Equations

Introduction: Systems of Linear Inequalities: A system of linear inequalities is a set of two or more linear inequalities in the same variables. The following example illustrates this, y < x + 2…………..Inequality 1 y ≥ 2x − 1…………Inequality 2 Solution of a System of Linear Inequalities: A solution of a system of linear inequalities […]