What two numbers have a sum of 14 and a difference of 4 ? Graph a system of equations to determine the two numbers.

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**.