Linear with Rationals

Grade 9 · algebra · 75 practice problems · read aloud

🔊 Listen to this explanation

Solving Linear Equations with Rational Numbers

🧠 What is it and Why is it Useful?

This is the process of finding the value of a variable (like x) in an equation where the coefficients or constants are fractions or decimals. Mastering this is crucial because real-world problems—from calculating recipes to budgeting—often involve rational numbers. It's the foundation for all future algebra work!

📝 Step-by-Step Guide

  1. Simplify Each Side: Use the distributive property and combine like terms on each side of the equation separately.
  2. Eliminate Denominators: Multiply every term on both sides by the Least Common Denominator (LCD) to clear fractions.
  3. Solve the Simpler Equation: Isolate the variable using inverse operations (add/subtract, then multiply/divide).
  4. Check Your Solution: Substitute your answer back into the original equation to verify it's correct.

🔍 Worked Examples

Example 1: Fractional Coefficients

Solve: (1/2)x + 3 = (3/4)x - 1

  1. LCD of 2 and 4 is 4. Multiply every term by 4:
    4*(1/2)x + 4*3 = 4*(3/4)x - 4*1 → 2x + 12 = 3x - 4
  2. Subtract 2x from both sides: 12 = x - 4
  3. Add 4 to both sides: x = 16

Example 2: Decimals

Solve: 0.5x - 1.2 = 0.2x + 0.3

  1. Multiply every term by 10 to clear decimals:
    10*0.5x - 10*1.2 = 10*0.2x + 10*0.3 → 5x - 12 = 2x + 3
  2. Subtract 2x: 3x - 12 = 3
  3. Add 12: 3x = 15
  4. Divide by 3: x = 5

⚠️ Common Mistakes to Avoid

  • Forgetting to multiply every term by the LCD: When clearing fractions, every single term on both sides must be multiplied.
  • Misapplying the distributive property: Remember a(b + c) = ab + ac, especially when the LCD is involved.
  • Sign errors with negatives: Be extra careful when moving terms across the equals sign. A negative sign belongs to the number that follows it.

💡 Tips & Tricks

  • LCD is your best friend! Clearing fractions first makes the equation look and feel much simpler.
  • Think of decimals as tenths, hundredths, etc. The multiplier 10, 100, etc., is just a special LCD.
  • Check with a calculator: After solving, plug your answer back into the original equation to catch calculation errors.

🎯 How to Practice

Start with simple equations with one fraction and gradually work up to more complex ones. Create your own problems and solve them. Use online practice sites like Khan Academy for instant feedback. The key is consistent, deliberate practice to build confidence and speed.

Practice problems

6 of the 75, worked through step by step — try them before opening the answer.

1 2x + 5 = 3(x - 1) + 7

Hint: Distribute any multiplication across parentheses first, then combine like terms on each side before isolating the variable.

Show the answer

Answer: x = 1

  1. Expand the right-hand side** 3(x - 1) means 3 times x and 3 times -1. So 3(x - 1) = 3x - 3. Now the equation becomes: 2x + 5 = 3x - 3 + 7 **
  2. Simplify the right-hand side** Combine -3 + 7 = 4. So: 2x + 5 = 3x + 4 **
  3. Move all terms with x to one side** Subtract 2x from both sides: 2x + 5 - 2x = 3x + 4 - 2x This simplifies to: 5 = x + 4 **
  4. Isolate x** Subtract 4 from both sides: 5 - 4 = x + 4 - 4 1 = x So the solution is: x = 1 **
  5. Check the solution** Plug x = 1 into the original equation: Left side: 2(1) + 5 = 2 + 5 = 7 Right side: 3(1 - 1) + 7 = 3(0) + 7 = 0 + 7 = 7 Both sides are equal, so x = 1 is correct. Final answer: x = 1

Let's solve the equation step-by-step. We start with: 2x + 5 = 3(x - 1) + 7 **

2 2(x - 3)/5 + (x + 4)/3 = 7

Hint: To solve equations with fractions, find a common denominator to eliminate them, then solve for the variable using inverse operations.

Show the answer

Answer: 11

  1. Multiply both sides by the least common denominator of 5 and 3, which is 15 15 × [2(x - 3)/5 + (x + 4)/3] = 15 × 7
  2. Distribute the 15 to each term 15 × 2(x - 3)/5 + 15 × (x + 4)/3 = 105
  3. Simplify each term 3 × 2(x - 3) + 5 × (x + 4) = 105 6(x - 3) + 5(x + 4) = 105
  4. Distribute the coefficients 6x - 18 + 5x + 20 = 105
  5. Combine like terms 11x + 2 = 105
  6. Subtract 2 from both sides 11x = 103
  7. Divide both sides by 11 x = 103/11 x = 11

The answer is 11.

3 (3x + 7)/2 - (2x - 5)/3 = 4

Hint: To solve equations with fractions, find a common denominator to eliminate them, then simplify and isolate the variable.

Show the answer

Answer: x = -1

When solving linear equations with rational coefficients, multiply both sides by the least common denominator to eliminate fractions. This transforms the equation into one with integer coefficients that can be solved using standard algebraic techniques like combining like terms and isolating the variable.

4 (2x - 5)/3 + (3x + 1)/4 = 7

Hint: To solve equations with fractions, find a common denominator to eliminate them first. For example, if you had (x+2)/5 + (x-1)/3 = 4, you would multiply through by 15.

Show the answer

Answer: 7

  1. Multiply both sides by the common denominator 12 to eliminate fractions 12 × [(2x - 5)/3 + (3x + 1)/4] = 12 × 7
  2. Distribute the 12 to each term 12 × (2x - 5)/3 + 12 × (3x + 1)/4 = 84
  3. Simplify each term 4(2x - 5) + 3(3x + 1) = 84
  4. Distribute the coefficients 8x - 20 + 9x + 3 = 84
  5. Combine like terms 17x - 17 = 84
  6. Add 17 to both sides 17x = 101
  7. Divide both sides by 17 x = 101/17
  8. Simplify the fraction x = 101 ÷ 17 = 7

The answer is 7.

5 3(2x - 5) + 4(x + 3) = 7x - 2

Hint: Distribute coefficients to eliminate parentheses, then combine like terms on each side before isolating the variable

Show the answer

Answer: x = 1

When solving linear equations with rational coefficients, first apply the distributive property to remove parentheses. Then combine all like terms on each side of the equation. Finally, use inverse operations to isolate the variable term on one side and the constant terms on the other side.

6 2(3x - 5) + 4(x + 2) = 7x + 12

Hint: Begin by distributing any coefficients outside parentheses, then combine like terms on each side before isolating the variable.

Show the answer

Answer: x = 14/3

  1. Distribute the numbers outside the parentheses** Multiply 2 by each term inside (3x - 5): 2 * 3x = 6x 2 * (-5) = -10 So 2(3x - 5) = 6x - 10 Multiply 4 by each term inside (x + 2): 4 * x = 4x 4 * 2 = 8 So 4(x + 2) = 4x + 8 Now the equation becomes: 6x - 10 + 4x + 8 = 7x + 12 --- **
  2. Combine like terms on the left side** 6x + 4x = 10x -10 + 8 = -2 So the left side simplifies to: 10x - 2 Now we have: 10x - 2 = 7x + 12 --- **
  3. Move all terms with x to one side and constants to the other** Subtract 7x from both sides: 10x - 7x - 2 = 12 3x - 2 = 12 Add 2 to both sides: 3x = 12 + 2 3x = 14 --- **
  4. Solve for x** Divide both sides by 3: x = 14/3 --- **Final answer:** x = 14/3

Let's solve the equation step by step. We start with: 2(3x - 5) + 4(x + 2) = 7x + 12 --- **

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