I am unsure of a few steps. Can someone help me?:)

Solve each system by the graphing method.

y = 1/2x + 1
4x - 8y = -8

a. Infinite solutions
b. (1, -8)
c. (1, 4)
d. No solution

2 Answers
Feb 22, 2018

a.) Infinite Solutions

Explanation:

graph{y=0.5x+1 [-10, 10, -5, 5]}
By graphing both on a plot, we can see that they are the same line. Therefore, there are infinite solutions to this.

If you wanted to do it analytically, you could solve the second equation for y:

4x-8y= -8 Subtract 4x over
-8y=-8color(red)(-4x) Divide both sides by -8
y=(-8-4x)/color(red)(-8) Simplify
y=1+x/2

This equation is equal to the first equation, so there will be infinite solutions.

Hope this helped!

Feb 22, 2018

Very detailed explanation given in the beginning using first principles to demonstrate where the shortcut methods come from.

Explanation:

Given:

y=1/2x+1" "..........Equation(1)
4x-8y=-8" ".....Equation(2)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

It is much easier to graph a straight line equation if it is in the form
y=mx+ c where m is the slope (gradient) and c is some constant value

Lets manipulate Eqn(2) into this format

color(blue)("Step 1 - Get the "y" term on its own")

We need to 'get rid of the 4x on the left. We do this by turning it into 0 as anything + 0 does not change.

subtract color(red)(4x) from ul("both sides.")

color(green)(4x-8ycolor(white)("d") =color(white)("d") -8color(white)("ddd")->color(white)("ddd")ubrace(4xcolor(red)(-4x))-8ycolor(white)("d")=color(white)("d")-8color(red)(-4x))

color(green)(color(white)("ddddddddddddddddd")->color(white)("dddd.d")0color(white)("dd")-8ycolor(white)("d")=-4x-8)

color(blue)("Step 2 - Get the "y" on its own")

We need to get rid of the 8 in -8y so turn it into 1 as 1xxy=y

Divide all of ul("both sides") by color(red)(8)

color(green)(-8ycolor(white)("d")=color(white)("d")-4x-8 color(white)("ddd")->color(white)("ddd")-8/color(red)(8) y=-4/color(red)(8)x-8/color(red)(8) )

color(green)(color(white)("ddddddddddddddddddd")->color(white)("dddd")-y=-1/2x-1 )

color(blue)("Step - Make the "y" positive")
Make y positive by multiplying everything ul("on both sides") by (-1) giving:

y=1/2x+1" ".....................Equation(2_a)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
color(magenta)("Spot anything?")

Notice that Eqn(1) and Eqn(2_a) are the same.

This means that Eqn(2_a) sits on top of Eqn(1)

So there is an infinite count of shared points
Tony BTony B