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What is a System of Equations?

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What is a System of Equations?

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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.

Slope Intercept Form of an Equation

y = mx + b

slope

y-intercept

Graphing a linear equation

y = 2x + 1

slope

y-intercept

y = 2x + 1

Assuming that all equations have one solution because those are the types of equations that are solved the most.

There are three types of solutions for systems of linear equations

One None Infinitely Many

y = x - 5

y = 4x +4

y = 2x + 1

y = x - 3

y = 4(x+1)

y = 3x + 1

Same Line infinitely many solutions

Parallel lines no solution

One point

one solution

Determine the number of solutions that the system has.

Slope: -4, y-intercept: -16

y = -4(x+4)

y = -4x-16

Slope: -4, y-intercept: -16

y = -4x-16

Determine the number of solutions that the system has.

Slope: 1, y-intercept: 1

y = x + 1

Slope: , y-intercept: 1

y = x + 1

Determine the number of solutions that the system has.

Slope: -5, y-intercept: 1

y = -5x + 1

Slope: -5, y-intercept: -2

y = -5x-2

How many solutions does the system of linear equations have?

y= -2x+1

y= -2x-2

How many solutions does the system of linear equations have?

y= -3x+1

y= -2x-7

How many solutions does the system of linear equations have?

y= -3(x+1)

y= -3x-3

- Create a second equation for the following system of equations so that they have one solution, no solution, and infinitely many solutions.

- One solution

- y = 7x + 1 y=

- No solution

- y = 7x + 1 y=

- Infinitely many solutions

- y = 7x + 1 y=

How many solutions does the system of equations have? y=6(x+1) and y=6x+6

a) 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-1

y = 2x-2

Not having the equation in slope- intercept form

y = 2 -x

3x + 2y = 4

y = -x + 2

-3x = -3x

2y = 4 -3x

Slope: -, y-intercept: 2

2 = 2

Graph the system of linear equations. Determine their solution.

y = x-1

y -2x = 2

+2x = +2x

y = 2 + 2x

y = 2x + 2

Graph the system of linear equations. Determine their solution.

Slope: 1, y-intercept: -1

y = x-1

Slope: 2, y-intercept: -2

y = 2x-2

y = x-1

0 = 1-1

0 = 0

(1,0)

y = 2x-2

0 = 2*1-2

0 = 0

Graph the system of linear equations. Determine their solution.

y = x+3

-4x + y = 0

+4x = +4x

y = 4x

Graph the system of linear equations. Determine their solution.

Slope: 4, y-intercept: 0

y = 4x

Slope: 1, y-intercept: 3

y = x+3

y = 4x

4 = 4*1

(1,4)

4 = 4

y = x+3

4 = 1+3

4 = 4

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

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

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

Graph the system of linear equations. Determine their solution.

y - 4 = 2x

y = 2x

+ 4 = + 4

y = 2x + 4

Graph the system of linear equations. Determine their solution.

Slope: 2, y-intercept: 0

y = 2x

Slope: 2, y-intercept: 4

y = 2x+4

y = 2x

y = 2x+4

Graph the system of linear equations. Determine their solution.

-x + 2y = 0

2y –x = - 2

+x = +x

+x = +x

2y = x - 2

2y = x

2 = 2

2 = 2

y = x- 1

y = x

Graph the system of linear equations. Determine their solution.

Slope: , y-intercept: 0

y = x

Slope: , y-intercept: -1

y = x- 1

y = x

y = x- 1

y = x

y = x- 1

Find the solution for the system of linear equations y=3x and y=3x-2 by graphing.

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

Graph the system of linear equations. Determine their solution.

y = 2(2x+4)

y – 4x = 8

y = 4x+8

+4x = +4x

y = 4x + 8

Graph the system of linear equations. Determine their solution.

Slope: 4, y-intercept: 8

y = 2(2x+4)

y = 4x+8

Slope: 4, y-intercept: 8

y = 4x+8

y = 2(2x+4)

y = 4x+8

y = 4x+8

Graph the system of linear equations. Determine their solution.

x -y = -1

y =x-2+3

-x =-x

y =x+1

-y = -1-x

-1 = -1

y= 1 +x

y= x + 1

Graph the system of linear equations. Determine their solution.

Slope:1, y-intercept: 1

y =x+1

Slope:1, y-intercept: 1

y =x-2+3

y =x+1

y =x+1

y =x-2+3

y =x+1

y-3x = 3

y =3(x+1)

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