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Linear Equations - ACT Math

  1. A contractor calculates materials with 4(x+2)=324(x + 2) = 32. Which value of xx satisfies 4(x+2)=324(x + 2) = 32?
    Answer: 66 This is a linear equation with parentheses. Starting with 4(x+2)=324(x + 2) = 32, divide both sides by 4 to get x+2=8x + 2 = 8. Subtract 2 from both sides to get x=6x = 6. The value of x is 66.
  2. What is the solution to the equation 2x−3=x+72x - 3 = x + 7?
    Answer: 10 This is a linear equation with variable terms on both sides. To solve 2x - 3 = x + 7, subtract x from both sides: x - 3 = 7. Add 3 to both sides: x = 10. The solution is x = 10.
  3. Solve for xx: 12x+4=8x+2812x + 4 = 8x + 28
    Answer: 6 This is a linear equation with variable terms on both sides. To solve 12x + 4 = 8x + 28, subtract 8x from both sides: 4x + 4 = 28. Subtract 4 from both sides: 4x = 24. Divide by 4: x = 6. The solution is x = 6.
  4. What is the value of xx if 5x+3=2x+125x + 3 = 2x + 12?
    Answer: 3 This is a linear equation with variable terms on both sides. To solve 5x + 3 = 2x + 12, subtract 2x from both sides: 3x + 3 = 12. Subtract 3 from both sides: 3x = 9. Divide by 3: x = 3. The solution is x = 3.
  5. Solve for xx: 5−2x=175 - 2x = 17.
    Answer: −6-6 This equation has a negative coefficient for x. Starting with 5 - 2x = 17, subtract 5 from both sides: -2x = 12. Divide by -2: x = -6. Remember that dividing by a negative number changes the sign.
  6. If  6(x+2)−3=21 \,6(x + 2) - 3 = 21\,, what is xx?
    Answer: 22 First distribute the 6: 6x + 12 - 3 = 21. Simplify: 6x + 9 = 21. Subtract 9 from both sides: 6x = 12. Divide by 6: x = 2. The value of x is 2.
  7. Solve for xx: 8x−6=2(x+3)8x - 6 = 2(x + 3)
    Answer: 2 This is a linear equation with parentheses on one side. To solve 8x−6=2(x+3)8x - 6 = 2(x + 3), first distribute the right side: 8x−6=2x+68x - 6 = 2x + 6. Subtract 2x from both sides: 6x−6=66x - 6 = 6. Add 6 to both sides: 6x=126x = 12. Divide by 6: x=2x = 2. The solution is x=2x = 2.
  8. If 4x+8=2x+144x + 8 = 2x + 14, what is xx?
    Answer: 3 This is a linear equation with variable terms on both sides. To solve 4x + 8 = 2x + 14, subtract 2x from both sides: 2x + 8 = 14. Subtract 8 from both sides: 2x = 6. Divide by 2: x = 3. The solution is x = 3.
  9. A music subscription discount is modeled by 5x−10=2x+85x - 10 = 2x + 8. Which value of xx satisfies the equation?
    Answer: 66 This is a linear equation with variables on both sides. Starting with 5x - 10 = 2x + 8, subtract 2x from both sides to get 3x - 10 = 8. Add 10 to both sides to get 3x = 18. Divide by 3 to get x = 6.
  10. A taxi ride costs 22 dollars per mile plus a 55-dollar base fee. If the total cost is $19, solve for xx in 2x+5=192x + 5 = 19.​​
    Answer: 77 This equation models a taxi fare with a base fee plus cost per mile. To solve 2x + 5 = 19, first subtract 5 from both sides: 2x = 14. Then divide by 2: x = 7. The ride was 7 miles.
  11. A phone plan includes a base charge and an extra fee. If the total is modeled by 4x−7=254x - 7 = 25, what is the value of xx?
    Answer: 8 This is a linear equation 4x - 7 = 25 modeling a phone plan total with base and extra fees, where x is the variable. Add 7 to both sides: 4x = 32. Divide both sides by 4: x = 8. Therefore, the value of x is 8, matching choice C. Choice B (6) might occur from subtracting 7 instead of adding, leading to 4x = 18 and x=4.5, but that's not a choice; precision in operations is essential.
  12. A class collects canned goods. The total is modeled by x+14=3x−2x + 14 = 3x - 2. What is the solution to the equation?
    Answer: 88 This is a linear equation with variables on both sides. Starting with x + 14 = 3x - 2, subtract x from both sides to get 14 = 2x - 2. Add 2 to both sides to get 16 = 2x. Divide by 2 to get x = 8.
  13. A hiker's elevation changes at a constant rate. If 2x−3=x+92x - 3 = x + 9, what is the value of xx?
    Answer: 12 This is a linear equation 2x - 3 = x + 9 representing elevation change, where x is the rate or related value. Subtract x from both sides: x - 3 = 9. Add 3 to both sides: x = 12. Therefore, the value of x is 12, matching choice C. Choice B (9) could come from subtracting 3 instead of adding, getting x = 6, but that's choice A; sign errors are common here.
  14. Two friends compare their savings. One has 3x+23x + 2 dollars and the other has x+14x + 14 dollars. If they have the same amount, solve for xx in 3x+2=x+143x + 2 = x + 14.​​
    Answer: 66 This equation has variables on both sides representing equal savings. To solve 3x + 2 = x + 14, subtract x from both sides: 2x + 2 = 14. Subtract 2: 2x = 12. Divide by 2: x = 6.
  15. A gym charges a one-time signup fee plus a monthly cost. If the total after 4 months is modeled by the equation 3x+8=203x + 8 = 20, what is the value of xx?
    Answer: 44 This is a linear equation where we need to isolate the variable x. Starting with 3x + 8 = 20, subtract 8 from both sides to get 3x = 12. Divide both sides by 3 to get x = 4. The value of x is 4.
  16. A recipe is adjusted by scaling ingredients. If 3(x−2)=213(x - 2) = 21, solve for xx.
    Answer: 9 This is a linear equation with distribution: 3(x - 2) = 21 adjusts a recipe by scaling, where x is the scaled amount. Distribute the 3: 3x - 6 = 21. Add 6 to both sides: 3x = 27. Divide by 3: x = 9. Therefore, the value of x is 9, which is choice C. Choice B (7) could come from forgetting to distribute and dividing 21 by 3 directly to get x - 2 = 7 without adding 2.
  17. If 7x=−567x = -56, what is xx?
    Answer: −8-8 This is a simple multiplication equation. To solve 7x = -56, divide both sides by 7: x = -56 ÷ 7 = -8. The negative sign must be preserved, so x = -8.
  18. What is the value of xx if 3x+6=153x + 6 = 15?
    Answer: 3 This is a two-step linear equation. To solve 3x + 6 = 15, first subtract 6 from both sides to get 3x = 9. Then divide both sides by 3 to isolate x: x = 3. The answer is 3.
  19. A student tracks points earned and lost in a game. If 9−2x=19 - 2x = 1, what is the value of xx?
    Answer: 4 This is a linear equation 9 - 2x = 1 tracking game points, where x represents points lost or a related variable. Subtract 9 from both sides: -2x = 1 - 9, so -2x = -8. Divide both sides by -2, remembering to flip the sign: x = 4. Therefore, the value of x is 4, matching choice C. Choice B (3) could occur from a sign mistake, like dividing -8 by 2 without the negative, but careful handling of negatives is crucial.
  20. A store sells a bundle of items. The cost is modeled by 4x−6=184x - 6 = 18. Which value of xx satisfies 4x−6=184x - 6 = 18?​​
    Answer: 66 This is a two-step linear equation. To solve 4x - 6 = 18, first add 6 to both sides: 4x = 24. Then divide by 4: x = 6. Choice A (x = 3) would give 4(3) - 6 = 6, not 18.

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