Exercise 117 Page 433

In this system of equations, at least one of the variables has a coefficient of 1. This means that we can approach its solution with the Equal Values Method. When solving a system of equations using this system, there are four steps.

1. Rewrite both equations into slope-intercept form, if necessary.
2. Set both expressions that are equal to y equal to each other to form a new equation.
3. Solve this equation for x.
4. Substitute this solution into one of the equations and solve for y. For this system, y is already isolated in both equations, so we can skip straight to solving!
   y=2x+1 & (I) y=- 3x-4 & (II)

   Substitute

   (II):y= 2x+1

   y=2x+1 2x+1=- 3x-4

   AddEqn

   (II): LHS+3x=RHS+3x

   y=2x+1 5x+1=- 4

   SubEqn

   (II): LHS-1=RHS-1

   y=2x+1 5x=- 5

   DivEqn

   (II): .LHS /5.=.RHS /5.

   y=2x+1 5x5= - 55

   SimpQuot

   (II): Simplify quotient

   y=2x+1 x= - 55

   CalcQuot

   (II): Calculate quotient

   y=2x+1 x=- 1
   Great! Now, to find the value of y, we need to substitute x=- 1 into either of the equations in the given system. Let's use the first equation.
   y=2x+1 x=- 1

   Substitute

   (I):x= - 1

   y=2( - 1)+1 x=- 1

   MultPosNeg

   (I): a(- b)=- a * b

   y=- 2+1 x=- 1

   AddTerms

   (I): Add terms

   y=- 1 x=- 1
   The solution, or point of intersection, to this system of equations is the point (- 1,- 1).

   Checking Our Answer

   Let's Check!

   To check our answer, we will substitute our solution into both equations. If doing so results in true statements, then our solution is correct.

   y=2x+1 & (I) y=- 3x-4 & (II)

   (I), (II): x= - 1, y= - 1

   - 1? =2( - 1)+1 - 1? =- 3( - 1)-4

   MultPosNeg

   (I): a(- b)=- a * b

   - 1? =- 2+1 - 1? =- 3(- 1)-4

   MultNegNegOnePar

   (II): - a(- b)=a* b

   - 1? =- 2+1 - 1? =3-4

   (I), (II): Add and subtract terms

   - 1=- 1 ✓ - 1=- 1 ✓

   Because both equations are true statements, we know that our solution is correct.