Exercise 29 Page 228

The ordered pair (- 9,1) is a solution to the given system of equations. Therefore, if we substitute x= - 9 and y= 1, the equations will still hold true. a( - 9)+b( 1)=- 31 a( - 9)-b( 1)=- 41 ⇔ - 9a+b=- 31 & (I) - 9a-b=- 41 & (II) Now we have an system with only a and b as variables. We will solve this system of equations for these variables using the Substitution Method. To do so, we will start by isolating the b-variable in Equation (I).

- 9a+b=- 31 - 9a-b=- 41

AddEqn

(I): LHS+9a=RHS+9a

b=- 31+9a - 9a-b=- 41

Let's now substitute - 31+9a for b in Equation (II).

b=- 31+9a - 9a-b=- 41

Substitute

(II): b= - 31+9a

b=- 31+9a - 9a-( - 31+9a)=- 41

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(II): Solve for a

Distr

(II): Distribute - 1

b=- 31+9a - 9a+31-9a=- 41

SubTerms

(II): Subtract terms

b=- 31+9a - 18a+31=- 41

SubEqn

(II): LHS-31=RHS-31

b=- 31+9a - 18a=- 72

DivEqn

(II): .LHS /(- 18).=.RHS /(- 18).

b=- 31+9a a=4

Finally, we will substitute a=4 in Equation (I), and simplify.

b=- 31+9a a=4

Substitute

(I): a= 4

b=- 31+9( 4) a=4

Multiply

(I): Multiply

b=- 31+36 a=4

AddTerms

(I): Add terms

b=5 a=4

We found that a= 4 and b=5. By substituting these values into the original equations, we obtain a linear system of equations with solution (- 9,1). 4x+5y=- 31 4x-5y=- 41