Exercise 21 Page 210

In this system of equations, at least one of the variables has a coefficient of 1. Therefore, we will approach its solution with the Substitution Method. When solving a system of equations using this method, there are three steps.

1. Isolate a variable in one of the equations.
2. Substitute the expression for that variable into the other equation and solve.
3. Substitute this solution into one of the equations and solve for the value of the other variable. Since the expression equal to y in the second equation is simpler, let's use that for our initial substitution.
   y=2x+5 & (I) y=- 3x & (II)

   Substitute

   (I):y= - 3x

   - 3x=2x+5 y=- 3x

   SubEqn

   (I): LHS-2x=RHS-2x

   - 3x-2x=2x+5-2x y=- 3x

   ▼

   (I): Subtract term

   CommutativePropAdd

   (I): Commutative Property of Addition

   - 3x-2x=2x-2x+5 y=- 3x

   SubTerm

   (I): Subtract term

   - 5x=5 y=- 3x

   DivEqn

   (I): .LHS /(- 5).=.RHS /(- 5).

   - 5x- 5= 5- 5 y=- 3x

   ▼

   (I): Calculate quotient

   MovePartNumRight

   (I): a* b/c=a/c* b

   - 5- 5x= 5- 5 y=- 3x

   MoveNegDenomToFrac

   (I): Put minus sign in front of fraction

   - 5- 5x=- 55 y=- 3x

   QuotOne

   (I): a/a=1

   1x=- 1 y=- 3x

   IdPropMult

   (I): Identity Property of Multiplication

   x=- 1 y=- 3x

   Great! Now, to find the value of y, we need to substitute x=- 1 into the second equation.

   x=- 1 y=- 3x

   Substitute

   (II):x= - 1

   x=- 1 y=- 3( - 1)

   MultNegNegOnePar

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

   x=- 1 y=3(1)

   IdPropMult

   (II): Identity Property of Multiplication

   x=- 1 y=3

   The solution, or point of intersection, to this system of equations is the point (- 1,3).

   Checking Our Answer

   Checking Our Answer

   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+5 & (I) y=- 3x & (II)

   (I), (II): x= - 1, y= 3

   3? =2( - 1)+5 3? =- 3( - 1)

   MultPosNeg

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

   3? =- 2(1)+5 3? =- 3(- 1)

   MultNegNegOnePar

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

   3? =- 2(1)+5 3? =3(1)

   (I), (II): Identity Property of Multiplication

   3? =- 2+5 3=3

   AddTerms

   (I): Add terms

   3=3 ✓ 3=3 ✓

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