Question

A pair of equations is shown below: y = 3x - 5 y = 6x - 8

Part A: Explain how you will solve the pair of equations by substitution or elimination. Show all the steps and write the solution. (5 points)

Part B: If the two equations are graphed, at what point will the lines representing the two equations intersect? Explain your answer. (5 points) ​

Answer 1

PART A

We start with: y = 3x - 5 y = 6x - 8

Elimination:

• If we have two equations that = y, then we can make them equal each other:
• y = y
• 3x - 5 = 6x - 8
• We then collect the x term together:
• 3x - 6x = -8 + 5
• -3x = -3
• We can divide both sides by -3:
• x = 1

Substitution

• We can make one equation turn from a y= to an x=
• I decided to go with y = 3x - 5 (but it doesn't matter)
• We basically want to get it into x=
• y = 3x - 5
• y + 5 = 3x
• 3x = y + 5 (I'm just flipping the equation to get x on the left for easiness)
• We then divide both sides by 3:
• x =
  `(y + 5)/(3)`
• Don't forget we've also got y = 6x - 8
• So we just put x in!
• y = 6 ×
  `(y + 5)/(3)` - 8
• y =
  `(6)/(3)` × (y + 5) - 8
• y = 2 × (y + 5) - 8
• y = 2y + 10 - 8
• y = 2y + 2
• Rearrange to get y on one side:
• -y = 2
• Times both sides by -1:
• y = -2
• We can now go back to...
• x =
  `(y + 5)/(3)`
• By putting y = -2 in:
• x =
  `(-2 + 5)/(3)`
• x =
  `(3)/(3)` = 1

PART B

• We can find where they intersect/overlap by using elimination/substiution
• We've already covered both, and we get x = 1
• This means we've already go the x-coordinate of our point where the lines meet
• (x, y) = (1, ?)
• We just put our value of x into either equation (since x = 1 is where they meet)
• I'll go with y = 6x - 8
• y = 6 × 1 - 8
• y = 6 - 8 = -2
• We end up with (1, -2)!!!

P.S.:

I've attached a screenshot of the graphs, notice how they meet at (1, -2)

Answer 2

See below

Explanation:

y = 3x -5 <===== multiply this equation by -2 to get

-2y = -6x+10 now add it to the other equation(this will ELIMINATE x)

y = 6x-8

-y = +2

y = -2 then sub this in to oneof the quations to find x = 1

the two points will intersect at x,y = 1,-2 (because this satisfies both of the equations)