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.

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: http://youtu.be/Br7qn4yLf-I

‘All I do is solve’ Rap: http://youtu.be/1qHTmxlaZWQ

2.2.2: Solving Systems of Linear Equations

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IntroductionThe 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

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Key Concepts, continued

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2.2.2: Solving Systems of Linear Equations

Intersecting Lines Parallel Lines Same Line

One solution No solutions Infinitely many solutions

ConsistentIndependent

Inconsistent ConsistentDependent

Guided Practice

Example 1Graph 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.

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2.2.2: Solving Systems of Linear Equations

Guided Practice: Example 1, continued

1. 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. 5

2.2.2: Solving Systems of Linear Equations

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).

Guided Practice: Example 1, continued

2. 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.

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2.2.2: Solving Systems of Linear Equations

Guided Practice: Example 1, continued

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2.2.2: Solving Systems of Linear Equations

3. Observe the graph.

The lines intersect at the point (3, 0).

This appears to be the solution to this system of equations.

Guided Practice: Example 1, continued

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2.2.2: Solving Systems of Linear Equations

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.

To check, substitute (3, 0) into both original equations. The result should be 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.

Guided Practice: Example 1, continued

4. 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.

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2.2.2: Solving Systems of Linear Equations

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Guided Practice

Example 2Graph 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.

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2.2.2: Solving Systems of Linear Equations

Guided Practice: Example 2, continued

1. Solve each equation for y.

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2.2.2: Solving Systems of Linear Equations

Guided Practice: Example 2, continued

2. Graph both equations using the slope-intercept method.

The y-intercept of both equations is 1.

The slope of both equations is 2.

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2.2.2: Solving Systems of Linear Equations

Guided Practice: Example 2, continued

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2.2.2: Solving Systems of Linear Equations

3. Observe the graph.The graphs of y = 2x +1 and -8x + 4y = 4 are thesame line.

There are infinitely manysolutions to this system ofequations.

Guided Practice: Example 2, continued

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2.2.2: Solving Systems of Linear Equations

Guided Practice: Example 2, continued

4. 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.

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2.2.2: Solving Systems of Linear Equations

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Guided Practice

Example 3Graph 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.

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2.2.2: Solving Systems of Linear Equations

Guided Practice: Example 3, continued

1. 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.

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2.2.2: Solving Systems of Linear Equations

–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).

Guided Practice: Example 3, continued

2. 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.

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2.2.2: Solving Systems of Linear Equations

Guided Practice: Example 3, continued

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2.2.2: Solving Systems of Linear Equations

3. Observe the graph.The graphs of –6x + 2y = 8and y = 3x – 5 are parallellines and never cross.

Guided Practice: Example 3, continued

4. 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.

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2.2.2: Solving Systems of Linear Equations

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