What is a System of Equations?

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Vocabulary

Can you look at a system of linear equations and tell how many

solutions it has?

For example:Could you tell that the equations y=2x +1 and y= 2x-7 have no

solution?

In this lesson you will learn to predict how many solutions a

system of linear equations has by inspection.

Let’s ReviewLet’s Review

Slope Intercept Form of an Equation

y = mx + b

slope y-intercept

Let’s ReviewLet’s Review

Graphing a linear equation

y = 2x + 1

slope y-intercept

Let’s ReviewLet’s Review

y = 2x + 1

Let’s ReviewA Common MistakeAssuming that all equations have one solution because those are the types of equations that are solved the most.

Let’s ReviewCore LessonThere are three types of solutions

for systems of linear equations

One None Infinitely Many

Let’s ReviewCore Lesson

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

One pointone solution

y = 4x +4

Same Line

infinitely many solutions

Parallel lines no solution

y = x - 5

y = x - 3 y = 4(x+1)

Let’s ReviewCore Lesson

Determine the number of solutions that the system has.

y = -4(x+4)

y = -4x-16

Slope: -4, y-intercept: -16

Slope: -4, y-intercept: -16y = -4x-16

Let’s ReviewCore Lesson

Determine the number of solutions that the system has.

y = x + 1

y = x + 1

Slope: 1, y-intercept: 1

Slope: , y-intercept: 1

Let’s ReviewCore Lesson

Determine the number of solutions that the system has.

y = -5x + 1

y = -5x-2

Slope: -5, y-intercept: 1

Slope: -5, y-intercept: -2

Let’s ReviewGuided Practice

How many solutions does the system of linear equations have?

y= -2x+1

y= -2x-2

Let’s ReviewGuided Practice

How many solutions does the system of linear equations have?

y= -3x+1

y= -2x-7

Let’s ReviewGuided Practice

How many solutions does the system of linear equations have?

y= -3(x+1)y= -3x-3

Let’s ReviewExtension ActivitiesCreate a second equation for the following system of equations so that they have one solution, no solution, and infinitely many solutions. y = 7x + 1 y= y = 7x + 1 y= y = 7x + 1 y=

One solutionNo solutionInfinitely many solutions

Let’s ReviewQuick QuizHow many solutions does the system of equations have? y=6(x+1) and y=6x+6a) One b) None c) Infinitely Many

How many solutions does the system of equations have? y=-3x and y=x+1 a) One b) None c) Infinitely Many

What happens if we graph a system of equations and the lines

intersect?

y = x-1y = 2x-2

Let’s ReviewA Common Mistake

Not having the equation in slope- intercept form

3x + 2y = 4-3x = -3x2y = 4 -3x

2 = 2

y = 2 - xy = - x + 2

Slope: - , y-intercept: 2

Let’s ReviewCore Lesson

Graph the system of linear equations. Determine their solution.

y = x-1 y -2x = 2 +2x =

+2xy = 2 + 2xy = 2x + 2

Let’s ReviewCore Lesson

Graph the system of linear equations. Determine their solution.

y = x-1

y = 2x-2

Slope: 1, y-intercept: -1

Slope: 2, y-intercept: -2

Let’s ReviewCore Lesson

y = x-1

y = 2x-2

0 = 1-10 = 0

0 = 2*1-20 = 0

(1,0)

Let’s ReviewCore Lesson

Graph the system of linear equations. Determine their solution.

-4x + y = 0 y = x+3 +4x = +4x y = 4x

Let’s ReviewCore Lesson

Graph the system of linear equations. Determine their solution.

y = 4x

y = x+3

Slope: 4, y-intercept: 0

Slope: 1, y-intercept: 3

Let’s ReviewCore Lesson

y = 4x

y = x+3

4 = 4*14 = 4

4 = 1+34 = 4

(1,4)

Let’s ReviewExtension Activities

When we solve the systems of equations y=2x and y=x, what is our solution? What does it mean?

Let’s ReviewExtension Activities

How many solutions does the system of equations y=2x+1 and y=x+1 have? Why?

Let’s ReviewQuick Quiz

The solution for the system of linear equations y=3x+4 and y=x+2 is?a) (-1,1) b) (1,-1) c)(-1,-1) d) (1,1)

The solution for the system of linear equations y=x+6 and y=-2x is?a) (2,-4) b) (-2,4) c) (-2,-4) d) (2,4)

What happens if we graph a system of equations and the lines are

parallel?

y = x

y = x - 1

Let’s ReviewCore Lesson

Graph the system of linear equations. Determine their solution.

y = 2x y - 4 = 2x + 4 = + 4y = 2x + 4

Let’s ReviewCore Lesson

Graph the system of linear equations. Determine their solution.

y = 2x

y = 2x+4

Slope: 2, y-intercept: 0

Slope: 2, y-intercept: 4

Let’s ReviewCore Lesson

y = 2x

y = 2x+4

Let’s ReviewCore Lesson

Graph the system of linear equations. Determine their solution.

-x + 2y = 0 2y –x = - 2 +x = +x 2y = x 2 = 2

y = x

+x = +x 2y = x - 2 2 = 2

y = x - 1

Let’s ReviewCore Lesson

Graph the system of linear equations. Determine their solution.

y = x

y = x - 1

Slope: , y-intercept: 0

Slope: , y-intercept: -1

Let’s ReviewCore Lesson

y = x - 1

y = x

Let’s ReviewCore Lesson

y = x - 1

y = x

Let’s ReviewGuided PracticeFind the solution for the system of linear equations y=3x and y=3x-2 by graphing.

Let’s ReviewQuick Quiz

The solution for the system of linear equations y=2x+1 and y=2x-3 is?a) (-1,1) b) (1,-1) c)(-1,-1) d) no solution

The solution for the system of linear equations y=-x+6 and y=-x-2 is?a) (2,-4) b) (-2,4) c) (-2,-4) d) no solution

What happens if we graph a system of equations and the lines are the

same?

y = 2(2x+4)y = 4x+8

Let’s ReviewCore Lesson

Graph the system of linear equations. Determine their solution.

y = 2(2x+4) y – 4x = 8y = 4x+8 +4x =

+4xy = 4x + 8

Let’s ReviewCore Lesson

Graph the system of linear equations. Determine their solution.

y = 2(2x+4)

y = 4x+8

Slope: 4, y-intercept: 8

Slope: 4, y-intercept: 8

y = 4x+8

Let’s ReviewCore Lesson

y = 2(2x+4)

y = 4x+8

y = 4x+8

Let’s ReviewCore Lesson

Graph the system of linear equations. Determine their solution.

x -y = -1 y =x-2+3y =x+1 -x =-x

-y = -1-x -1 = -1 y= 1 +x y= x + 1

Let’s ReviewCore Lesson

Graph the system of linear equations. Determine their solution.

y =x+1

y =x-2+3

Slope:1, y-intercept: 1

Slope:1, y-intercept: 1y =x+1

Let’s ReviewCore Lesson

y =x-2+3

y =x+1

y =x+1

Let’s ReviewGuided Practice

y =3(x+1)

y-3x = 3

Let’s ReviewQuick Quiz

The solution for the system of linear equations y=2x+1 and y=2x-3 is?a) One solution b)Infinitely Many Solutions

The solution for the system of linear equations y=4x-12and y=4(x+3)is?a) One solution b) Infinitely Many Solutions