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Solving Equations with Variables on Both Sides 11-3
IWBAT solve equations with variables on
both sides of the equal sign.
Solving Equations with Variables on Both Sides 11-3
WRITE:
Some problems produce equations that have variables on both sides of the equal sign.
Solving an equation with variables on both sides is similar to solving an equation with a variable on only one side. You can add or subtract a term containing a variable on both sides of an equation.
Solving Equations with Variables on Both Sides 11-3
Solve.
4x + 6 = x
EX 1A: Solving Equations with Variables on Both
Sides
4x + 6 = x
– 4x – 4x
6 = –3x
Subtract 4x from both sides.
Divide both sides by –3.
–2 = x
6 –3
–3x –3
=
Solving Equations with Variables on Both Sides 11-3
Check your solution by substituting the value back into the original equation. For example, 4(-2) + 6 = -2 or -2 = -2.
Helpful Hint
WRITE:
Solving Equations with Variables on Both Sides 11-3
Solve.
9b – 6 = 5b + 18
EX 1B: Solving Equations with Variables on Both
Sides
9b – 6 = 5b + 18
– 5b – 5b
4b – 6 = 18
4b 4
24 4
=
Subtract 5b from both sides.
Divide both sides by 4.
b = 6
+ 6 + 6
4b = 24
Add 6 to both sides.
Solving Equations with Variables on Both Sides 11-3
Solve.
9w + 3 = 9w + 7
EX 1C: Solving Equations with Variables on Both
Sides
3 ≠ 7
9w + 3 = 9w + 7
– 9w – 9w Subtract 9w from both sides.
No solution. There is no number that can be substituted for the variable w to make the equation true.
Solving Equations with Variables on Both Sides 11-3
If the variables in an equation are eliminated and the resulting statement is false, the equation has no solution.
Helpful Hint
WRITE:
Solving Equations with Variables on Both Sides 11-3
To solve multi-step equations with variables on both sides, first combine like terms and clear fractions. Then add or subtract variable terms to both sides so that the variable occurs on only one side of the equation. Then use properties of equality to isolate the variable.
WRITE:
Solving Equations with Variables on Both Sides 11-3
Solve.
10z – 15 – 4z = 8 – 2z - 15
EX 2: Solving Multi-Step Equations with Variables on
Both Sides
10z – 15 – 4z = 8 – 2z – 15
+ 15 +15
6z – 15 = –2z – 7 Combine like terms.
+ 2z + 2z Add 2z to both sides.
8z – 15 = – 7
8z = 8
z = 1
Add 15 to both sides.
Divide both sides by 8. 8z 8 8 8
=
Solving Equations with Variables on Both Sides 11-3
EX 3: Business Application
Daisy’s Flowers sell a rose bouquet for $39.95 plus $2.95 for every rose. A competing florist sells a similar bouquet for $26.00 plus $4.50 for every rose. Find the number of roses that would make both florists’ bouquets cost the same price.
Solving Equations with Variables on Both Sides 11-3
EX 3 Continued
39.95 + 2.95r = 26.00 + 4.50r Let r represent the price of one rose.
– 2.95r – 2.95r
39.95 = 26.00 + 1.55r
Subtract 2.95r from both sides.
– 26.00 – 26.00 Subtract 26.00 from both sides.
13.95 = 1.55r
13.95 1.55
1.55r 1.55
= Divide both sides by 1.55.
9 = r The two services would cost the same when purchasing 9 roses.
Solving Equations with Variables on Both Sides 11-3
EX 4: Multi-Step Application
Jamie spends the same amount of money each morning. On Sunday, he bought a newspaper for $1.25 and also bought two doughnuts. On Monday, he bought a newspaper for fifty cents and bought five doughnuts. On Tuesday, he spent the same amount of money and bought just doughnuts. How many doughnuts did he buy on Tuesday?
Solving Equations with Variables on Both Sides 11-3
EX 4 Continued
First solve for the price of one doughnut.
1.25 + 2d = 0.50 + 5d Let d represent the price of one doughnut.
– 2d – 2d
1.25 = 0.50 + 3d
Subtract 2d from both sides.
– 0.50 – 0.50 Subtract 0.50 from both sides.
0.75 = 3d
0.75 3
3d 3
= Divide both sides by 3.
0.25 = d The price of one doughnut is $0.25.
Solving Equations with Variables on Both Sides 11-3
EX 4 Continued
Now find the amount of money Jamie spends each morning.
1.25 + 2d Choose one of the original expressions.
Jamie spends $1.75 each morning.
1.25 + 2(0.25) = 1.75
0.25n 0.25
1.75 0.25
=
Let n represent the number of doughnuts.
Find the number of doughnuts Jamie buys on Tuesday.
0.25n = 1.75
n = 7; Jamie bought 7 doughnuts on Tuesday.
Divide both sides by 0.25.
Solving Equations with Variables on Both Sides 11-3
EX 5: Solving Literal Equations for a Variable
The equation t = m + 10e gives the test score t for a student who answers m multiple-choice questions and e essay questions correctly. Solve this equation for e.
t = m + 10e Locate e in the equation.
Since m is added to 10e,
subtract m from both sides.
t = m + 10e
–m –m
t – m = 10e
t – m = 10e
10 10 t – m = e
10
Since e is multiplied 10, divide
both sides by 10.
