💡 6th Grade Math: Two-Step Equations Practice Questions
1
Solved Example
Easy Level
Solve the following equation for \(x\):
\[ 2x + 5 = 15 \]
💡 Goal: Isolate the variable \(x\) by performing inverse operations.
Solution & Explanation
Here's how to solve for \(x\):
Step 1: Subtract 5 from both sides of the equation to isolate the term with \(x\).
\(2x + 5 - 5 = 15 - 5\)
\(2x = 10\)
Step 2: Divide both sides by 2 to solve for \(x\).
\(\frac{2x}{2} = \frac{10}{2}\)
\(x = 5\)
✅ Solution: \(x = 5\)
2
Solved Example
Easy Level
Find the value of \(y\) in the equation:
\[ 3y - 7 = 8 \]
👉 Remember to perform operations in the reverse order of operations (PEMDAS/BODMAS).
Solution & Explanation
Let's solve for \(y\):
Step 1: Add 7 to both sides of the equation.
\(3y - 7 + 7 = 8 + 7\)
\(3y = 15\)
Step 2: Divide both sides by 3.
\(\frac{3y}{3} = \frac{15}{3}\)
\(y = 5\)
✅ Solution: \(y = 5\)
3
Solved Example
Medium Level
What is the value of \(a\) in this equation?
\[ \frac{a}{4} + 2 = 6 \]
📌 Tip: Division is the inverse of multiplication, and subtraction is the inverse of addition.
Solution & Explanation
Solving for \(a\):
Step 1: Subtract 2 from both sides.
\(\frac{a}{4} + 2 - 2 = 6 - 2\)
\(\frac{a}{4} = 4\)
Step 2: Multiply both sides by 4 to isolate \(a\).
\(\frac{a}{4} \times 4 = 4 \times 4\)
\(a = 16\)
✅ Solution: \(a = 16\)
4
Solved Example
Medium Level
Solve for \(b\):
\[ 5b - 10 = 20 \]
💡 Think: What is the first step to get the term with \(b\) by itself?
Solution & Explanation
Let's find \(b\):
Step 1: Add 10 to both sides.
\(5b - 10 + 10 = 20 + 10\)
\(5b = 30\)
Step 2: Divide both sides by 5.
\(\frac{5b}{5} = \frac{30}{5}\)
\(b = 6\)
✅ Solution: \(b = 6\)
5
Solved Example
Medium Level
Sarah bought \(m\) markers for 2 each and a notebook for 3. She spent a total of 11. Write and solve a two-step equation to find out how many markers Sarah bought.
👉 Hint: The total cost is the cost of markers plus the cost of the notebook.
Solution & Explanation
Let's set up and solve the equation:
Step 1: Define the variable. Let \(m\) be the number of markers.
Step 2: Write the equation. The cost of \(m\) markers is \(2m\). The total cost is \(2m + 3\). We know this total is 11.
\[ 2m + 3 = 11 \]
Step 3: Solve the equation. Subtract 3 from both sides.
\(2m + 3 - 3 = 11 - 3\)
\(2m = 8\)
Step 4: Divide both sides by 2.
\(\frac{2m}{2} = \frac{8}{2}\)
\(m = 4\)
✅ Solution: Sarah bought 4 markers.
6
Solved Example
Medium Level
John had \(c\) cookies. He gave 5 cookies to his friend and then ate half of the remaining cookies. If he has 7 cookies left, how many cookies did John start with?
💡 Reverse the steps: Think about what happened last and work backward.
Solution & Explanation
Let's solve this step-by-step:
Step 1: Let \(c\) be the number of cookies John started with.
Step 2: After giving 5 away, he had \(c - 5\) cookies.
Step 3: He ate half of the remaining, so he had \(\frac{c - 5}{2}\) cookies left.
Step 4: We know he has 7 cookies left, so the equation is:
\[ \frac{c - 5}{2} = 7 \]
Step 5: Solve the equation. Multiply both sides by 2.
\(\frac{c - 5}{2} \times 2 = 7 \times 2\)
\(c - 5 = 14\)
Step 6: Add 5 to both sides.
\(c - 5 + 5 = 14 + 5\)
\(c = 19\)
✅ Solution: John started with 19 cookies.
7
Solved Example
Real World Example
You are saving up for a new video game that costs 60. You already have 15 saved. You decide to save 5 each week. How many weeks will it take you to save enough money?
👉 Think: What is the total amount needed? What do you already have? How much are you saving per week?
Solution & Explanation
Let's figure out how many weeks it will take:
Step 1: Let \(w\) be the number of weeks.
Step 2: The amount saved from weeks is \(5w\).
Step 3: The total amount saved will be your initial savings plus the weekly savings: \(15 + 5w\).
Step 4: Set this equal to the cost of the game:
\[ 15 + 5w = 60 \]
Step 5: Solve the equation. Subtract 15 from both sides.
\(15 + 5w - 15 = 60 - 15\)
\(5w = 45\)
Step 6: Divide both sides by 5.
\(\frac{5w}{5} = \frac{45}{5}\)
\(w = 9\)
✅ Solution: It will take 9 weeks to save enough money.
8
Solved Example
Real World Example
A baker made 3 batches of cupcakes. He used \(x\) cups of flour for each batch and had 2 cups of flour left over. If he used a total of 14 cups of flour, how many cups of flour were in each batch?
📌 Remember: The total flour used is the sum of flour from all batches.
Solution & Explanation
Let's find out how much flour was in each batch:
Step 1: Let \(x\) be the number of cups of flour per batch.
Step 2: The total flour used for 3 batches is \(3x\).
Step 3: The equation representing the total flour used is \(3x + 2\).
Step 4: We know the total flour used was 14 cups:
\[ 3x + 2 = 14 \]
Step 5: Solve the equation. Subtract 2 from both sides.
\(3x + 2 - 2 = 14 - 2\)
\(3x = 12\)
Step 6: Divide both sides by 3.
\(\frac{3x}{3} = \frac{12}{3}\)
\(x = 4\)
✅ Solution: There were 4 cups of flour in each batch.
6th Grade Math: Two-Step Equations Practice Questions
Example 1:
Solve the following equation for \(x\):
\[ 2x + 5 = 15 \]
💡 Goal: Isolate the variable \(x\) by performing inverse operations.
Solution:
Here's how to solve for \(x\):
Step 1: Subtract 5 from both sides of the equation to isolate the term with \(x\).
\(2x + 5 - 5 = 15 - 5\)
\(2x = 10\)
Step 2: Divide both sides by 2 to solve for \(x\).
\(\frac{2x}{2} = \frac{10}{2}\)
\(x = 5\)
✅ Solution: \(x = 5\)
Example 2:
Find the value of \(y\) in the equation:
\[ 3y - 7 = 8 \]
👉 Remember to perform operations in the reverse order of operations (PEMDAS/BODMAS).
Solution:
Let's solve for \(y\):
Step 1: Add 7 to both sides of the equation.
\(3y - 7 + 7 = 8 + 7\)
\(3y = 15\)
Step 2: Divide both sides by 3.
\(\frac{3y}{3} = \frac{15}{3}\)
\(y = 5\)
✅ Solution: \(y = 5\)
Example 3:
What is the value of \(a\) in this equation?
\[ \frac{a}{4} + 2 = 6 \]
📌 Tip: Division is the inverse of multiplication, and subtraction is the inverse of addition.
Solution:
Solving for \(a\):
Step 1: Subtract 2 from both sides.
\(\frac{a}{4} + 2 - 2 = 6 - 2\)
\(\frac{a}{4} = 4\)
Step 2: Multiply both sides by 4 to isolate \(a\).
\(\frac{a}{4} \times 4 = 4 \times 4\)
\(a = 16\)
✅ Solution: \(a = 16\)
Example 4:
Solve for \(b\):
\[ 5b - 10 = 20 \]
💡 Think: What is the first step to get the term with \(b\) by itself?
Solution:
Let's find \(b\):
Step 1: Add 10 to both sides.
\(5b - 10 + 10 = 20 + 10\)
\(5b = 30\)
Step 2: Divide both sides by 5.
\(\frac{5b}{5} = \frac{30}{5}\)
\(b = 6\)
✅ Solution: \(b = 6\)
Example 5:
Sarah bought \(m\) markers for 2 each and a notebook for 3. She spent a total of 11. Write and solve a two-step equation to find out how many markers Sarah bought.
👉 Hint: The total cost is the cost of markers plus the cost of the notebook.
Solution:
Let's set up and solve the equation:
Step 1: Define the variable. Let \(m\) be the number of markers.
Step 2: Write the equation. The cost of \(m\) markers is \(2m\). The total cost is \(2m + 3\). We know this total is 11.
\[ 2m + 3 = 11 \]
Step 3: Solve the equation. Subtract 3 from both sides.
\(2m + 3 - 3 = 11 - 3\)
\(2m = 8\)
Step 4: Divide both sides by 2.
\(\frac{2m}{2} = \frac{8}{2}\)
\(m = 4\)
✅ Solution: Sarah bought 4 markers.
Example 6:
John had \(c\) cookies. He gave 5 cookies to his friend and then ate half of the remaining cookies. If he has 7 cookies left, how many cookies did John start with?
💡 Reverse the steps: Think about what happened last and work backward.
Solution:
Let's solve this step-by-step:
Step 1: Let \(c\) be the number of cookies John started with.
Step 2: After giving 5 away, he had \(c - 5\) cookies.
Step 3: He ate half of the remaining, so he had \(\frac{c - 5}{2}\) cookies left.
Step 4: We know he has 7 cookies left, so the equation is:
\[ \frac{c - 5}{2} = 7 \]
Step 5: Solve the equation. Multiply both sides by 2.
\(\frac{c - 5}{2} \times 2 = 7 \times 2\)
\(c - 5 = 14\)
Step 6: Add 5 to both sides.
\(c - 5 + 5 = 14 + 5\)
\(c = 19\)
✅ Solution: John started with 19 cookies.
Example 7:
You are saving up for a new video game that costs 60. You already have 15 saved. You decide to save 5 each week. How many weeks will it take you to save enough money?
👉 Think: What is the total amount needed? What do you already have? How much are you saving per week?
Solution:
Let's figure out how many weeks it will take:
Step 1: Let \(w\) be the number of weeks.
Step 2: The amount saved from weeks is \(5w\).
Step 3: The total amount saved will be your initial savings plus the weekly savings: \(15 + 5w\).
Step 4: Set this equal to the cost of the game:
\[ 15 + 5w = 60 \]
Step 5: Solve the equation. Subtract 15 from both sides.
\(15 + 5w - 15 = 60 - 15\)
\(5w = 45\)
Step 6: Divide both sides by 5.
\(\frac{5w}{5} = \frac{45}{5}\)
\(w = 9\)
✅ Solution: It will take 9 weeks to save enough money.
Example 8:
A baker made 3 batches of cupcakes. He used \(x\) cups of flour for each batch and had 2 cups of flour left over. If he used a total of 14 cups of flour, how many cups of flour were in each batch?
📌 Remember: The total flour used is the sum of flour from all batches.
Solution:
Let's find out how much flour was in each batch:
Step 1: Let \(x\) be the number of cups of flour per batch.
Step 2: The total flour used for 3 batches is \(3x\).
Step 3: The equation representing the total flour used is \(3x + 2\).
Step 4: We know the total flour used was 14 cups:
\[ 3x + 2 = 14 \]
Step 5: Solve the equation. Subtract 2 from both sides.
\(3x + 2 - 2 = 14 - 2\)
\(3x = 12\)
Step 6: Divide both sides by 3.
\(\frac{3x}{3} = \frac{12}{3}\)
\(x = 4\)
✅ Solution: There were 4 cups of flour in each batch.