Two times a number plus three times another equals 13. The sum of the two numbers is 7. What are the numbers?

Question

UpStudy AI · Accepted Answer

Let's solve the problem step by step.

**Problem Statement:**
- **Equation 1:** Two times a number plus three times another equals 13.
  
  $$ 2x + 3y = 13 $$

- **Equation 2:** The sum of the two numbers is 7.
  
  $$ x + y = 7 $$

Here, $$ x $$ and $$ y $$ represent the two unknown numbers we need to find.

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**Step 1: Express one variable in terms of the other using Equation 2**

From Equation 2:

$$ x + y = 7 $$

We can solve for $$ x $$:

$$ x = 7 - y $$

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**Step 2: Substitute the expression for $$ x $$ into Equation 1**

Substitute $$ x = 7 - y $$ into Equation 1:

$$ 2(7 - y) + 3y = 13 $$

**Simplify the equation:**

$$ 14 - 2y + 3y = 13 $$

Combine like terms:

$$ 14 + y = 13 $$

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**Step 3: Solve for $$ y $$**

Subtract 14 from both sides:

$$ y = 13 - 14 $$
$$ y = -1 $$

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**Step 4: Substitute $$ y = -1 $$ back into the expression for $$ x $$**

Using $$ x = 7 - y $$:

$$ x = 7 - (-1) $$
$$ x = 7 + 1 $$
$$ x = 8 $$

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**Solution:**
- The first number ($$ x $$) is **8**.
- The second number ($$ y $$) is **-1**.

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**Verification:**

Let's check if these values satisfy both original equations.

1. **Equation 1:**

$$ 2x + 3y = 2(8) + 3(-1) = 16 - 3 = 13 $$

$$ 13 = 13 $$ ✅

2. **Equation 2:**

$$ x + y = 8 + (-1) = 7 $$

$$ 7 = 7 $$ ✅

Both equations are satisfied, so the solution is correct.

**Final Answer:**  
The two numbers are **8** and **-1**.
The two numbers are 8 and -1.

Real Tutor Solution