1. Linear Equations by Graphing
Lesson 2.7
Solving Systems of
Linear Equations by Graphing
Concept: Solving systems of equations in two variables.
EQ: How can I manipulate equations to solve a system of equations? (REI 5-6,10-11)
Vocabulary: System of equations, Inconsistent, Consistent (independent/dependent)
‘In Common’ Ballad:
‘All I do is solve’ Rap:
2.2.2: Solving Systems of Linear Equations

2. Introduction
The solution to a system of equations is the point or points that make both equations true. Systems of equations can have one solution, no solutions, or an infinite number of solutions.
There are various methods to solving a system of equations. One is the graphing method.
On a graph, the solution to a system of equations can be easily seen. The solution to the system is the point of intersection, the point at which two lines cross or meet.
2.2.2: Solving Systems of Linear Equations

3. Key Concepts, continued
Intersecting Lines
Parallel Lines
Same Line
One solution
No solutions
Infinitely many solutions
Consistent
Independent
Inconsistent
Dependent
2.2.2: Solving Systems of Linear Equations

4. Guided Practice Example 1
Graph the system of equations. Then determine whether the system has one solution, no solution, or infinitely many solutions. If the system has a solution, name it.
2.2.2: Solving Systems of Linear Equations

5. Guided Practice: Example 1, continued Solve each equation for y.
The first equation needs to be solved for y.
The second equation (y = –x + 3) is already in slope-intercept form.
4x – 6y = 12
Original equation
–6y = 12 – 4x
Subtract 4x from both sides.
Divide both sides by –6.
Write the equation in slope-intercept form (y = mx + b).
2.2.2: Solving Systems of Linear Equations

6. Guided Practice: Example 1, continued
Graph both equations using the slope-intercept method.
The y-intercept of is –2. The slope is .
The y-intercept of y = –x + 3 is 3. The slope is –1.
2.2.2: Solving Systems of Linear Equations

7. Guided Practice: Example 1, continued
3. Observe the graph.
The lines intersect at the point (3, 0).
This appears to be the solution to this system of equations.
2.2.2: Solving Systems of Linear Equations

8. Guided Practice: Example 1, continued
To check, substitute (3, 0) into both original equations. The result should be a true statement.
4x – 6y = 12
First equation in the system
4(3) – 6(0) = 12
Substitute (3, 0) for x and y.
12 – 0 = 12
Simplify.
12 = 12
This is a true statement.
y = –x + 3
Second equation in the system
(0) = –(3) + 3
Substitute (3, 0) for x and y.
0 = –3 + 3
Simplify.
0 = 0
This is a true statement.
2.2.2: Solving Systems of Linear Equations

9. ✔ Guided Practice: Example 1, continued The system has one
solution, (3, 0).
This system is consistent because it has at least one solution and it is independent because it only has 1 solution. ✔
2.2.2: Solving Systems of Linear Equations

10. Guided Practice Example 2
Graph the system of equations. Then determine whether the system has one solution, no solution, or infinitely many solutions. If the system has a solution, name it.
2.2.2: Solving Systems of Linear Equations

11. Guided Practice: Example 2, continued Solve each equation for y.
2.2.2: Solving Systems of Linear Equations

12. Guided Practice: Example 2, continued
Graph both equations using the slope-intercept method.
The y-intercept of both equations is 1.
The slope of both equations is 2.
2.2.2: Solving Systems of Linear Equations

13. Guided Practice: Example 2, continued
Observe the graph.
The graphs of y = 2x +1
and -8x + 4y = 4 are the
same line.
There are infinitely many
solutions to this system of
equations.
2.2.2: Solving Systems of Linear Equations

14. Guided Practice: Example 2, continued
2.2.2: Solving Systems of Linear Equations

15. ✔ Guided Practice: Example 2, continued The system has infinitely
many solutions.
This system is consistent because it has at least one solution and it is dependent because it has more than one solution. ✔
2.2.2: Solving Systems of Linear Equations

16. Guided Practice Example 3
Graph the system of equations. Then determine whether the system has one solution, no solution, or infinitely many solutions. If the system has a solution, name it.
2.2.2: Solving Systems of Linear Equations

17. Guided Practice: Example 3, continued Solve each equation for y.
The first equation needs to be solved for y.
The second equation (y = 3x – 5) is already in slope-intercept form.
–6x + 2y = 8
Original equation
2y = 8 + 6x
Add 6x to both sides.
y = 4 + 3x
Divide both sides by 2.
y = 3x + 4
Write the equation in slope-intercept form (y = mx + b).
2.2.2: Solving Systems of Linear Equations

18. Guided Practice: Example 3, continued
Graph both equations using the slope-intercept method.
The y-intercept of y = 3x + 4 is 4. The slope is 3.
The y-intercept of y = 3x – 5 is –5. The slope is 3.
2.2.2: Solving Systems of Linear Equations

19. Guided Practice: Example 3, continued
Observe the graph.
The graphs of –6x + 2y = 8
and y = 3x – 5 are parallel
lines and never cross.
2.2.2: Solving Systems of Linear Equations

20. ✔ Guided Practice: Example 3, continued The system has no solutions
because there are no values for x and y that will make both equations true. This system is inconsistent because it has no solutions. ✔
2.2.2: Solving Systems of Linear Equations