Rational Coefficient Equations

Grade 7 · algebra · 94 practice problems · read aloud

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What Are Rational Coefficient Equations? 🤔

These are equations where the variable (like x) is multiplied by a fraction. For example, (2/3)x = 10. They are super useful because they appear everywhere—in recipes, calculating distances, and splitting up groups fairly!

How to Solve Them: A Step-by-Step Guide

  1. Identify the coefficient: Find the fraction in front of your variable.
  2. Use the reciprocal: Multiply both sides of the equation by the reciprocal (flipped fraction) of the coefficient.
  3. Simplify: Multiply the fractions and simplify your answer.
  4. Check your solution: Plug your answer back into the original equation to make sure it works!

Worked Examples

Example 1: (3/4)x = 12

Step 1: The coefficient is 3/4.

Step 2: Multiply both sides by the reciprocal, 4/3.
(4/3) * (3/4)x = 12 * (4/3)

Step 3: Simplify.
(12/12)x = 48/3
1x = 16
x = 16

Example 2: (2/5)y = 7

Step 1: Coefficient is 2/5.

Step 2: Multiply both sides by 5/2.
(5/2) * (2/5)y = 7 * (5/2)

Step 3: Simplify.
1y = 35/2
y = 35/2 or 17.5

Common Mistakes to Avoid ⚠️

Forgetting to multiply both sides: Whatever you do to one side of the equation, you MUST do to the other to keep it balanced.

Misplacing the negative sign: If your coefficient is negative, like -1/2, its reciprocal is also negative (-2/1). The negative sign flips with the fraction.

Not simplifying: Always simplify your final fraction answer if possible.

Tips & Tricks

Reciprocal Rule: Remember, a number multiplied by its reciprocal always equals 1. This is the magic that makes the variable isolated!

Cross-Multiply Shortcut: For a simple equation like (a/b)x = c, you can think of it as x = (c * b) / a. This is just a faster way of using the reciprocal.

How to Practice

  • Start with simple fractions like 1/2, 1/3, 1/4.
  • Create your own problems and solve them.
  • Ask a friend or parent to give you numbers to make equations for you to solve.
  • Look for word problems that involve fractions—they often turn into these types of equations!

Practice problems

6 of the 94, worked through step by step — try them before opening the answer.

1 (3/4)x - 7 = 5

Hint: To solve equations with rational coefficients, first isolate the variable term by performing inverse operations on both sides of the equation.

Show the answer

Answer: 16

  1. Isolate the term with x by adding 7 to both sides. We do this because -7 is on the left side, and we want to remove it. (3/4)x - 7 + 7 = 5 + 7 This simplifies to: (3/4)x = 12
  2. Solve for x by getting rid of the fraction (3/4). Since x is multiplied by 3/4, we can multiply both sides by the reciprocal of 3/4, which is 4/3. So: (3/4)x * (4/3) = 12 * (4/3)
  3. Simplify both sides. On the left: (3/4)*(4/3) = 1, so we have x. On the right: 12 * (4/3) = (12/1)*(4/3) = (12*4)/(1*3) = 48/3 = 16 Therefore: x = 16 We can check: (3/4)*16 - 7 = (48/4) - 7 = 12 - 7 = 5, which matches the original equation.

We start with the equation: (3/4)x - 7 = 5

2 (3/4)x + 5 = 17

Hint: Isolate the variable term by moving the constant to the other side, then eliminate the fractional coefficient.

Show the answer

Answer: 16

  1. Subtract 5 from both sides to isolate the term with x. (3/4)x + 5 - 5 = 17 - 5 (3/4)x = 12
  2. To solve for x, we need to get rid of the fraction 3/4. Since (3/4)x means 3/4 multiplied by x, we can multiply both sides by the reciprocal of 3/4, which is 4/3. So multiply both sides by 4/3: x = 12 * (4/3)
  3. Simplify the right-hand side. 12 * (4/3) = (12/1) * (4/3) = (12 * 4) / (1 * 3) = 48 / 3 = 16
  4. Conclusion. x = 16 The solution is x = 16.

We are solving the equation: (3/4)x + 5 = 17

3 (3/4)x - 7 = 11

Hint: Isolate the variable term by performing inverse operations on both sides of the equation, then eliminate the fractional coefficient.

Show the answer

Answer: 24

  1. Isolate the term with x by adding 7 to both sides. We do this because -7 is being subtracted from (3/4)x, so the opposite operation is to add 7. (3/4)x - 7 + 7 = 11 + 7 This simplifies to: (3/4)x = 18
  2. Solve for x by getting rid of the fraction 3/4. Since x is multiplied by 3/4, we do the opposite operation, which is multiplying by the reciprocal of 3/4. The reciprocal of 3/4 is 4/3. Multiply both sides by 4/3: (4/3) * (3/4)x = 18 * (4/3)
  3. Simplify both sides. On the left: (4/3) * (3/4) = 1, so we have 1*x or just x. On the right: 18 * (4/3) = (18/1) * (4/3) = (18 * 4) / (1 * 3) = 72 / 3 = 24 So: x = 24
  4. Check the solution (optional but good practice). Substitute x = 24 into the original equation: (3/4)*24 - 7 = (72/4) - 7 = 18 - 7 = 11 This matches the original equation's right-hand side, so the solution is correct. Final answer: x = 24

We are solving the equation: (3/4)x - 7 = 11

4 (3/4)x - 7 = 5 = ?

Hint: When solving equations with fractional coefficients, first isolate the variable term by performing inverse operations on both sides of the equation.

Show the answer

Answer: 16

  1. Isolate the term with x. Add 7 to both sides to remove -7 from the left side. (3/4)x - 7 + 7 = 5 + 7 (3/4)x = 12
  2. Solve for x. Since x is multiplied by 3/4, we can multiply both sides by the reciprocal of 3/4, which is 4/3. x = 12 * (4/3)
  3. Simplify. 12 * 4 = 48 48 / 3 = 16 So, x = 16.

We start with the equation: (3/4)x - 7 = 5

5 3/4 × (2/3 + 1/6) = ?

Hint: Remember to perform operations inside parentheses first, then multiply the result by the fraction outside.

Show the answer

Answer: 5/8

  1. Simplify inside the parentheses** We need to add 2/3 and 1/6. First, find a common denominator. The denominators are 3 and 6. The least common denominator is 6. Convert 2/3 to a fraction with denominator 6: 2/3 = (2 × 2)/(3 × 2) = 4/6. Now add: 4/6 + 1/6 = (4 + 1)/6 = 5/6. So the expression becomes: 3/4 × (5/6) --- **
  2. Multiply the fractions** Multiply numerators: 3 × 5 = 15 Multiply denominators: 4 × 6 = 24 So we have: 15/24 --- **
  3. Simplify the fraction** Find the greatest common divisor (GCD) of 15 and 24. Factors of 15: 1, 3, 5, 15 Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 GCD is 3. Divide numerator and denominator by 3: 15 ÷ 3 = 5 24 ÷ 3 = 8 So 15/24 simplifies to 5/8. --- **Final Answer:** 5/8

Let's solve the problem step by step. We have: 3/4 × (2/3 + 1/6) --- **

6 3/4 × 16 + 2/3 × 15 = ?

Hint: When solving expressions with multiple operations, remember to perform multiplication before addition. Multiply each fraction by its corresponding whole number first, then add the results together.

Show the answer

Answer: 22

  1. Calculate 3/4 × 16** 3/4 × 16 means (3 × 16) / 4. 3 × 16 = 48 48 / 4 = 12 So, 3/4 × 16 = 12 --- **
  2. Calculate 2/3 × 15** 2/3 × 15 means (2 × 15) / 3. 2 × 15 = 30 30 / 3 = 10 So, 2/3 × 15 = 10 --- **
  3. Add the results** 12 + 10 = 22 --- **Final Answer:** 22

Let's solve step by step. We have: 3/4 × 16 + 2/3 × 15 --- **

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