GED Math Linear Equations and Systems Summary

1. Add 8 to both sides: 5x = 35.
2. Divide both sides by 5: x = 7.

Check: 5(7) − 8 = 35 − 8 = 27.

Wrong choices

• x = 3.8: x = 3.8 subtracts 8 instead of adding it: (27 − 8) ÷ 5.
• x = 35: x = 35 is the value of 5x; it still has to be divided by 5.
• x = 5.4: x = 5.4 divides 27 by 5 and ignores the −8.

1. Subtract 3 from both sides: x/4 = 7.
2. Multiply both sides by 4: x = 28.

Check: 28 ÷ 4 + 3 = 7 + 3 = 10.

Wrong choices

• x = 52: x = 52 adds 3 instead of subtracting it: (10 + 3) × 4.
• x = 7: x = 7 is the value of x/4, not x.
• x = 1.75: x = 1.75 divides 7 by 4 instead of multiplying.

1. Distribute: 3x − 12 = 2x + 5.
2. Subtract 2x from both sides: x − 12 = 5.
3. Add 12 to both sides: x = 17.

Check: Left side: 3(17 − 4) = 3 × 13 = 39. Right side: 2(17) + 5 = 39.

Wrong choices

• x = 9: x = 9 multiplies only the x by 3 and leaves −4 as is.
• x = −7: x = −7 moves the −12 to the other side without changing its sign.
• x = 3.4: x = 3.4 adds 2x to 3x instead of subtracting it.

1. Equation: 3 + 2.50m = 28.
2. Subtract 3: 2.50m = 25.
3. Divide by 2.50: m = 10 miles.

Check: $3.00 + 10 × $2.50 = $3.00 + $25.00 = $28.00.

Wrong choices

• 11.2 miles: 11.2 miles divides $28 by $2.50 and forgets to take out the $3.00 start fee.
• 12.4 miles: 12.4 miles adds the $3.00 fee instead of subtracting it.
• 25 miles: 25 is the dollar amount for the miles; it still has to be divided by $2.50.

1. She starts with $240.
2. Each week adds $35, so after w weeks she has added 35w.
3. Total: 240 + 35w = 590.

Check: Solving gives 35w = 350, so w = 10 weeks. 240 + 35 × 10 = 590.

Wrong choices

• 35 + 240w = 590: 35 + 240w = 590 adds $240 each week, which switches the two amounts.
• 240w + 35w = 590: 240w + 35w = 590 multiplies her starting savings by the weeks, but she has the $240 only once.
• 590 + 35w = 240: 590 + 35w = 240 starts with the goal and ends with the savings, which is backward.

1. Replace y with 2x in the second equation: x + 2x = 21.
2. Combine: 3x = 21.
3. Divide: x = 7.

Check: y = 2(7) = 14, and 7 + 14 = 21.

Wrong choices

• x = 10.5: x = 10.5 divides 21 by 2, as if x and y were equal.
• x = 14: x = 14 is the value of y, not x.
• x = 21: x = 21 is the sum of x and y, not x alone.

1. Add 5 to both sides: (2/3)x = 12.
2. Multiply both sides by 3/2: x = 12 × 3/2 = 18.

Check: (2/3)(18) − 5 = 12 − 5 = 7.

Wrong choices

• x = 8: x = 8 multiplies 12 by 2/3 again instead of by 3/2.
• x = 3: x = 3 subtracts 5 instead of adding it: (7 − 5) × 3/2.
• x = 12: x = 12 is the value of (2/3)x, not x.

1. Let a = adult tickets. Child tickets = 50 − a.
2. Money equation: 12a + 7(50 − a) = 500.
3. Distribute: 12a + 350 − 7a = 500.
4. Combine: 5a + 350 = 500, so 5a = 150 and a = 30.

Check: 30 adults × $12 = $360 and 20 children × $7 = $140. $360 + $140 = $500.

Wrong choices

• 20: 20 is the number of child tickets.
• 25: 25 adult tickets would bring in 25 × $12 + 25 × $7 = $475, not $500.
• 35: 35 adult tickets would bring in 35 × $12 + 15 × $7 = $525, not $500.

1. Subtract 0.4x from both sides: 2.6 = 0.5x − 1.4.
2. Add 1.4 to both sides: 4.0 = 0.5x.
3. Divide by 0.5: x = 8.

Check: Left: 0.4(8) + 2.6 = 3.2 + 2.6 = 5.8. Right: 0.9(8) − 1.4 = 7.2 − 1.4 = 5.8.

Wrong choices

• x = 2.4: x = 2.4 subtracts 1.4 instead of adding it: (2.6 − 1.4) ÷ 0.5.
• x = −8: x = −8 has a sign error; with x = −8 the left side is −0.6 and the right side is −8.6.
• x = 4: x = 4 is the value of 0.5x; it still has to be divided by 0.5.

1. Start: P = 2l + 2w.
2. Subtract 2l from both sides: P − 2l = 2w.
3. Divide both sides by 2: w = (P − 2l) / 2.

Check: For a 10 by 4 rectangle, P = 28. (28 − 2 × 10) ÷ 2 = 8 ÷ 2 = 4, the width.

Wrong choices

• w = P − l: w = P − l drops both 2s; with P = 28 and l = 10 it gives 18, not 4.
• w = (P − l) / 2: w = (P − l) / 2 subtracts only one length; it gives 9 instead of 4.
• w = (P + 2l) / 2: w = (P + 2l) / 2 adds 2l instead of subtracting it.