Name:Date:Period:

Topic: Solving Systems of Equations by GraphingEssential Question: How can graphing systems of equation help you find a solution?

Warm-Up: Vocabulary Match-up (Review)1. Equation a) same variable and exponent

2. Like terms b) intercept & slope is

opposite reciprocal

of each other

3. Parallel Lines c) uses an equal sign4. Perpendicular Lines d) where a graph crosses the y-axis

5. y – intercept e) never intercept &

same slope

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What is a system of equations?

A system of equations is when you have two or more equations using the same variables.

The solution to the system is the point that satisfies ALL of the equations. This point will be an ordered pair.

When graphing, you will encounter three possibilities.

Intersecting Lines

The point where the lines intersect is your solution.

What is the solution of these lines?

Parallel Lines

These lines never intersect!

Since the lines never cross, there is NO SOLUTION!

Parallel lines have the same slope with different y-intercepts.

2Slope = = 2

1y-intercept = 2

y-intercept = -1

Coinciding Lines

These lines are the same! Since the lines are on top

of each other, there are INFINITELY MANY SOLUTIONS!

Coinciding lines have the same slope and y-intercepts.

2Slope = = 2

1y-intercept = -1

What is the solution of the system graphed below?

1. (2, -2)

2. (-2, 2)

3. No solution

4. Infinitely many solutions

1) Find the solution to the following system by graphing:

2x + y = 4

x - y = 2To find the solution we must graph both equations.

- I will first graph using x- and y-intercepts (plug in zeros) and then I’ll graph using the slope-intercept form. Choose the one that you are more comfortable using.

Graph the equations.

2x + y = 4

x - y = 2

Where do the lines intersect?

Check your answer!

To check your answer, plug the point back into both equations.

2x + y = 4 2(2) + (0) = 4

x - y = 2(2) – (0) = 2 Nice job…let’s try another!

Answer to question # 1)

2x + y = 4

x - y = 2

Using x- and y-intercepts (plug in zeros).

2x + y = 4(0, 4) and (2, 0)

x – y = 2(0, -2) and (2, 0)

Answer: (2, 0)

2) Find the solution to the following system by graphing:

y = 2x – 3

-2x + y = 1Graph both equations. Put both equations in same form

(standard or slope-intercept form). I’ll start with slope-intercept form on this one!

Graphing the Systems of equations:

y = 2x – 3

-2x + y = 1

Answer to question # 2)

y = 2x – 3m = 2 and b = -3

y = 2x + 1m = 2 and b = 1

Where do the lines intersect?No solution!

Notice that the slopes are the same with different y-intercepts. If you recognize this early, you don’t

have to graph them!

3) What is the solution of this system?

3x – y = 82y = 6x -16

1. (3, 1)

2. (4, 4)

3. No solution

4. Infinitely many solutions

Answer to question # 3)

3x – y = 82y = 6x -16

•Put both equations in slope-intercept form:

3x – y = 83x – y = 8 --- To isolate for ‘y,’ first move ‘mx’ (slope & ‘x’ coordinate to the other side– y = – 3x + 8 --- When moving ‘mx’ to the other side of the equation change the symboly = 3x – 8 --- ‘y’ cannot be a negative value, so we divided everything by negative

2y = 6x -162y = 6x -16 --- To isolate for ‘y,’ divide everything by 2y = 3x - 8

Page 363 - 364 (10, 22, 26)

Graph the system of equations. Determine whether the system has one solution, no solution, or infinitely many solutions. If the system has one solution, determine the solution.

1 3 3

3 9 9

.

x y

x y

23

54

5 3

. y x

y x

3 3

2 6

. x y

x y

Writing a System of Equations:Challenge:

A plant nursery is growing a tree that is 3ft tall and grows at an average rate of 1 ft per year. Another

tree at the nursery is 4ft tall and grows at an average rate of 0.5ft per year. After how

many years will the trees be the same height?

Solving a system of equations by graphing.

Let's summarize! There are 3 steps to solving a system using a graph.

Step 1: Graph both equations.

Step 2: Do the graphs intersect?

Step 3: Check your solution.

Graph using slope– intercept form or x- and y-intercepts. Be sure to use a ruler and graph paper!

This is the solution! LABEL the solution!

Substitute the x and y values into both equations to verify the point is a solution to both equations.

Activity: Name the Quadrant

Materials:

• Color Pencils

• Rulers

• Graphing Paper

• Lined Paper

1. y = x – 1 y = 2x

2. y = 2x + 1 y = 3x

3. y = 5xy = -x – 6

4. y = 3x + 8 y = 3x + 4

10. -6 +y = 2x

y = x – 3

1 + 1

23

= + 72

y x

y x

9. 7 + y = 9x -9x + y = – 7

8. -2x + y = 6 y = 4x + 3

7. y = -x + 1 y = 2x + 7

3 -

21

+ 32

y x

y x

Activity: Name the Quadrant

5.

6.

Home-Learning Assignment #6

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