Solving System of Linear Equations by Graphing

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  àTrue

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

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 to 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 of Equations by Graphing.

2x – y = 6 à Equation 1

x + 3y = 10 à Equation 2

Solution:

First, graph 2x – y = 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      àTrue

Second equation:

4 + 3(2) = 4 + 6 = 10    àTrue

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 to the system.

2x – 5y = – 2 

3x + 4y = 4

3. Why is the point of intersection for a system of equations considered its solution?

4. Solve the system by graphing.

y = 2x + 5 àEquation 1

y = −4x − 1 àEquation 2

5. Solve the system by graphing.

y = x – 1 àEquation 1

y = −x + 3 àEquation 2

6. Use a graph to solve the following system of equations.

3x + 2y = 9

2/3y = 3 – x

7. Use a graph to solve the following system of equations.

y = 1/2x + 7

4x – 8y = 12

8. Use a graph to solve the following system of equations.

y = x

y = 2x + 1

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

10. 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.