## 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 2*x* – *y* = 6.

Second, graph *x* + 3*y* = 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

- A ______________________ consists of two or more linear equations.
- Determine whether (4, 2) is a solution of the system.
- 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.

### Concept Map

### What have we learned

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

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