Exercise 34 Page 229

Systems of equations can be solved using three methods — substitution, elimination, or by graphing. If we are not asked to use a particular method, we should choose the method that seems to be the easiest for us based on the system we want to solve. Let's take a look at the given system. 5x+2y=12 & (I) y=x-8 & (II) Equation 1 is in standard form and Equation 2 is in slope-intercept form. To solve this system by graphing we would probably like to rewrite Equation 1 into slope-intercept form because it is easier to find the coordinates of points. Therefore, solving by graphing would require 6 steps.

1. Rewrite Equation 1 into slope-intercept form.
2. Make a table of values for Equation 1.
3. Graph Equation 1.
4. Make a table of values for Equation 2.
5. Graph Equation 2.
6. Find the point of intersection of the lines.

However, since Equation 2 is already solved for y we could substitute the right-hand side for y into Equation 1. Solving this system by substitution will require 4 steps.

1. Substitute for y into Equation 1.
2. Solve Equation 1 for x.
3. Substitute the value of x into Equation 2.
4. Solve Equation 2 for y.

Therefore, it seems like using substitution requires less steps, so we should use this method instead of graphing.

Extra

Solving the System Using Substitution

Let's solve the system according to the steps we wrote down. First we will substitute the right-hand side of Equation 2 for y in Equation 1.

5x+2y=12 & (I) y=x-8 & (II)

Substitute

(I): y= x-8

5x+2( x-8)=12 y=x-8

Distr

(I): Distribute 2

5x+2* x-2* 8=12 y=x-8

Multiply

(I): Multiply

5x+2x-16=12 y=x-8

AddTerms

(I): Add terms

7x-16=12 y=x-8

AddEqn

(I): LHS+16=RHS+16

7x=28 y=x-8

DivEqn

(I): .LHS /7.=.RHS /7.

x=4 y=x-8

The value of x is equal to 4. Let's substitute this value into Equation 2.

x=4 y=x-8

Substitute

(II):x= 4

x=4 y= 4-8

SubTerm

(II): Subtract term

x=4 y=-4

The solution to the equation is (4,-4).