GED Mathematical Reasoning
Linear Equations
An equation is a balance scale with an equals sign in the middle. Whatever you do to one side, you must do to the other, or the scale tips and the equation stops being true. Solving means peeling everything off the variable until it sits alone.
The words first
- variable
- the letter standing in for the number you are hunting for
- solve
- get the variable alone on one side
- inverse operation
- the undo button — subtraction undoes addition, division undoes multiplication
- substitute
- put your answer back into the original to test it
The method
PEEL IN REVERSE ORDER
Undo + and − FIRST.
Undo × and ÷ SECOND.
(This is the opposite of the order you would use to calculate.)
3x + 7 = 22
3x + 7 − 7 = 22 − 7 ← take 7 off BOTH sides
3x = 15
3x ÷ 3 = 15 ÷ 3 ← divide BOTH sides by 3
x = 5
VARIABLES ON BOTH SIDES — gather them on one side first
5x − 4 = 2x + 11
5x − 2x − 4 = 11 ← subtract 2x from both sides
3x − 4 = 11
3x = 15
x = 5
WORD PROBLEMS WITH A FLAT FEE
total = flat fee + (rate × how many)
Subtract the flat fee FIRST, then divide by the rate.
Worked all the way through
The problem
A phone plan costs $30.00 per month plus $0.10 per minute. One month's bill was $64.00. How many minutes were used?
- Step 1 — Name what you are looking for.Minutes. Call it m.
- Step 2 — Translate the sentence into an equation.You pay 30 no matter what, plus a dime for each minute, and it added up to 64: 30 + 0.10m = 64
- Step 3 — Get the flat fee out of the way. Subtract 30 from BOTH sides.30 + 0.10m − 30 = 64 − 30 → 0.10m = 34
- Step 4 — Now undo the multiplication. Divide BOTH sides by 0.10.0.10m ÷ 0.10 = 34 ÷ 0.10 → m = 340
- Step 5 — Check it by substituting back.30 + 0.10 × 340 = 30 + 34 = 64 ✓ matches the bill
On the board
30 + 0.10m = 64
flat fee + rate × minutes = bill
30 + 0.10m − 30 = 64 − 30
take 30 off BOTH sides
0.10m = 34
0.10m ÷ 0.10 = 34 ÷ 0.10
divide BOTH sides by 0.10
m = 340
30 + 0.10(340) = 64 ✓
check by putting it back
Answer
340 minutes
A second one, in a different form
Solve 2x + 6 = 12.
2x + 6 = 12
the problem
2x = 12 − 6
take 6 off both sides
2x = 6
x = 6/2
divide both sides by 2
x = 3
2(3) + 6 = 12 ✓
check it
Answer
x = 3
The usual mistake
Dividing before clearing the added or subtracted term.
Check yourself
The trap is dividing 64 by 0.10 before subtracting the $30. The flat fee is not charged per minute, so it has to come off first.
Practice it
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