how do I graph y=x and y=4x-9? where do they intersect?

Question

Algebra 1

Solving systems of equations by graphing

Y=x

Y=4x-9

Answer 1

Equation of a Line

y=mx+b

m=slope [rise(up) / run(to the right)]

b=y intercept (point on the line where x=0)

 

Line 1

 

y=x is the same as y=1x+0

m=1 (rise 1 / run 1)

b=0

 

You can graph this equation by using two points to create a line.

 

Point 1: y intercept (0,0)

Point 2: start at the y intercept, use the slope to rise up 1 and run over 1 to the right, (0+1, 0+1) ending at (1,1)

 

Connect the two points to create a line.

 

To find other points on the line y=x, plug in values of x and solve for y.

 

y=x

When x=-1

y=-1

 

So another point on the line is (-1, -1).

 

You can continue to plug in values of x and solve for the corresponding value of y to find other points on the line. Use a ruler to connect all those points to create a line.

 

Do the same to graph the next equation.

 

Line 2

 

y=4x-9

m=4 (rise 4, run 1)

b=-9

 

Point 1: y intercept (0,-9)

Point 2: start at the y intercept, use the slope to rise up 4 and run over 1 to the right, (0+1,-9+4), ending at (1,-5)

 

Connect the two points to create the second line. To find other points on the line y=4x-9, plug in values of x and solve for y.

 

y=4x-9

When x=-1

y=4(-1)-9

y=-4-9

y=-13

 

So, another point on the line is (-1,-13). 

 

You can continue to plug in values of x and solve for the corresponding value of y to find other points on the line. Use a ruler to connect all those points to create a line.

 

Intersection of Line 1 and Line 2

 

 

The two lines intersect where they are equal to one another.

 

y=x

y=4x-9

 

4x-9=x

4x-9(+9)=x(+9) [add 9 to both sides]

4x=x+9

4x(-x)=x+9(-x) [subtract x from both sides]

3x=9

3x(/3)=9(/3) [divide both sides by 3]

x=3

 

So the equations intersect when x=3. To find the point where the two lines intersect, you need to plug x=3 back into the equations to solve for the y value when x=3. When you plug x=3 back into your equations, the y values should be the same for both equations because the two lines intersect at one point.

 

Line 1

 

y=x

y=3

 

Line 2

 

y=4x-9

y=4(3)-9

y=12-9

y=3

 

The two lines intersect at (3,3).