What is a System of Equations? Use the iPads to find information about:  Uses  Ways to solve  Real-world applications  Types  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 Review Slope Intercept Form of an Equation y = mx + b slopey-intercept

Graphing a linear equation y = 2x + 1 slopey-intercept

y = 2x + 1

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

Let’s Review Core Lesson There are three types of solutions for systems of linear equations One None Infinitely Many

Let’s Review Core Lesson y = 2x + 1 y = 3x + 1 One point one solution y = 4x +4 Same Line infinitely many solutions Parallel lines no solution y = 4(x+1)

Let’s Review Core 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: -16 y = -4x-16

Let’s Review Core Lesson Determine the number of solutions that the system has. y = x + 1 Slope: 1, y-intercept: 1

Let’s Review Core 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 Review Guided Practice How many solutions does the system of linear equations have? y= -2x+1 y= -2x-2

Let’s Review Guided Practice How many solutions does the system of linear equations have? y= -3x+1 y= -2x-7

Let’s Review Guided Practice How many solutions does the system of linear equations have? y= -3(x+1) y= -3x-3

Let’s Review Extension Activities Create 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 solution No solution Infinitely many solutions

Let’s Review Quick Quiz 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

Let’s Review A Common Mistake Not having the equation in slope- intercept form 3x + 2y = 4 -3x = -3x 2y = 4 -3x 2 = 2

Let’s Review Core Lesson Graph the system of linear equations. Determine their solution. y = x-1y -2x = 2 +2x = +2x y = 2 + 2x y = 2x + 2

Let’s Review Core 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 Review Core Lesson y = x-1 y = 2x-2 0 = = 0 0 = 2*1-2 0 = 0 (1,0)

Let’s Review Core Lesson Graph the system of linear equations. Determine their solution. -4x + y = 0 y = x+3 +4x = +4x y = 4x

Let’s Review Core 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 Review Core Lesson y = 4x y = x+3 4 = 4*1 4 = 4 4 = = 4 (1,4)

Let’s Review Extension Activities When we solve the systems of equations y=2x and y=x, what is our solution? What does it mean?

Let’s Review Extension Activities How many solutions does the system of equations y=2x+1 and y=x+1 have? Why?

Let’s Review Quick 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?

Let’s Review Core Lesson Graph the system of linear equations. Determine their solution. y = 2x y - 4 = 2x + 4 = + 4 y = 2x + 4

Let’s Review Core 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 Review Core Lesson y = 2x y = 2x+4

Let’s Review Core Lesson Graph the system of linear equations. Determine their solution. -x + 2y = 02y – x = - 2 +x = +x 2y = x 2 = 2 +x = +x 2y = x = 2

Let’s Review Core Lesson Graph the system of linear equations. Determine their solution.

Let’s Review Core Lesson

Let’s Review Core Lesson

Let’s Review Guided Practice Find the solution for the system of linear equations y=3x and y=3x-2 by graphing.

Let’s Review Quick 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 Review Core Lesson Graph the system of linear equations. Determine their solution. y = 2(2x+4)y – 4x = 8 y = 4x+8 +4x = +4x y = 4x + 8

Let’s Review Core 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 Review Core Lesson y = 2(2x+4) y = 4x+8 y = 4x+8

Let’s Review Core Lesson Graph the system of linear equations. Determine their solution. x -y = -1 y =x-2+3 y =x+1 -x =-x -y = -1-x -1 = -1 y= 1 +x y= x + 1

Let’s Review Core 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: 1 y =x+1

Let’s Review Core Lesson y =x-2+3 y =x+1 y =x+1

Let’s Review Guided Practice y =3(x+1) y-3x = 3

Let’s Review Quick 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