Simplify expressions to solve equations.
When solving equations, we should simplify the expressions that make up the left and
right sides before applying any properties of equality. Often, that involves removing
parentheses and/or combining like terms.
28.4 More about Solving Equations 671EXAMPLE (^5) Solve: a. b.
StrategyWe will use the distributive property along with the process of
combining like terms to simplify the left side of each equation.
WHYIt’s best to simplify each side of an equation before using a property of
equality.
Solution
a. This is the equation to solve.
Distribute the multiplication by 3.
On the right side, simplify:.
Check: This is the original equation.
Substitute for.
Do the addition within the parentheses. Think of 1
as and then add:.
Do the multiplications.
True
The solution is.
Caution! To check a result, we evaluate each side of the equation following
the order of operations rule. For the check shown above, perform the addition
within parentheses first.Don’t distribute the multiplication by 3.
Add first
b. This is the equation to solve.
Distribute the multiplication by 2.
Combine like terms: 8a 2 a 6 a. First,
we want to isolate the variable term, 6a.
To undo the addition of 14, subtract 14 from
both sides. This isolates 6a.
6 a 14 14 68 14
6 a 14 68
8 a 2 a 14 68
8 a 2 (a7) 68
⎧⎪⎨⎪⎩
3 a3
2
1 b3
2
0 0
15
2
15
2
0
3 a 22 32 22 (^52)
5
2
b 5 a3
2
b 03 a 32 k3
2
1 b 5 a3
2
b 03(k1) 5 k 0k ^32 233
2
To undo the multiplication by 2, divide both(^) sides by 2. This isolates k.
2 k
2
3
2
Do the subtractions: 3 3 0 and 0 3 3. 2 k (^3) Now we want to isolate the variable, k.
To undo the addition of 3, subtract 3 from both sides.
2 k 3 3 0 (^3) This isolates 2 k.
Combine like terms: 3k 5 k 2 k.
2 k 3 (^0) First, we want to isolate the variable term, 2 k.
3 k 3 5 k 0
3(k1) 5 k 0
3(k1) 5 k 0 8 a2(a7) 68
Self Check 5
Solve:
a.
b.
Now TryProblems 39 and 45
9 x5(x9) 1
4(a2)a 11
68
14
54