Answer
The two numbers are 9 and 5.
Solution
To find the two numbers with a **sum of 14** and a **difference of 4**, we can set up and solve a system of equations. Additionally, I'll explain how to graph these equations to visualize the solution.
### Step 1: Define the Variables
Let’s denote the two numbers as:
- **\( x \)**: The first number
- **\( y \)**: The second number
### Step 2: Set Up the Equations
Based on the problem statement, we have two conditions:
1. **Sum of the numbers is 14**:
\[
x + y = 14 \quad \text{(Equation 1)}
\]
2. **Difference of the numbers is 4**:
\[
x - y = 4 \quad \text{(Equation 2)}
\]
### Step 3: Solve the System of Equations
**Method 1: Addition Method**
Add **Equation 1** and **Equation 2** to eliminate \( y \):
\[
\begin{align*}
(x + y) + (x - y) &= 14 + 4 \\
2x &= 18 \\
x &= \frac{18}{2} \\
x &= 9
\end{align*}
\]
Now, substitute \( x = 9 \) back into **Equation 1** to find \( y \):
\[
\begin{align*}
9 + y &= 14 \\
y &= 14 - 9 \\
y &= 5
\end{align*}
\]
**Solution:** The two numbers are **9** and **5**.
### Step 4: Graphing the System of Equations
To graph the system of equations, follow these steps:
1. **Rewrite Both Equations in Slope-Intercept Form (\( y = mx + b \)):**
- **Equation 1:** \( x + y = 14 \)
\[
y = -x + 14
\]
- **Equation 2:** \( x - y = 4 \)
\[
y = x - 4
\]
2. **Plot the Lines on a Graph:**
- **For \( y = -x + 14 \):**
- **Y-intercept (0,14):** When \( x = 0 \), \( y = 14 \)
- **X-intercept (14,0):** When \( y = 0 \), \( x = 14 \)
- Draw a straight line passing through (0,14) and (14,0).
- **For \( y = x - 4 \):**
- **Y-intercept (0,-4):** When \( x = 0 \), \( y = -4 \)
- **Another Point:** When \( x = 8 \), \( y = 4 \) (since \( 8 - 4 = 4 \))
- Draw a straight line passing through (0,-4) and (8,4).
3. **Find the Intersection Point:**
The two lines intersect at the point where both equations are satisfied simultaneously. From our earlier solution, we know this is at \( (9,5) \).
*(Note: Since I can't create images, imagine two lines crossing at the point (9,5))*
### Conclusion
By solving the system of equations algebraically and understanding the graphical representation, we've determined that the two numbers are **9** and **5**.
Reviewed and approved by the UpStudy tutoring team
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