The sum of two numbers is 75
 and their difference is 15
. What are the two numbers? Let x=
 one number and y=
 the other number. The corresponding modeling system is {x+y=75x−y=15
. Solve the system graphically.

Answer
Keyboard Shortcuts
Graph the linear equations by writing the equations in slope-intercept form:

 Equation 1 slope
  
 Equation 1 intercept
 
 Equation 2 slope
  
 Equation 2 intercept
 
Identify the appropriate number of solutions. If there is a solution, give the point:

 
 first coordinate
  
 second coordinate
 
 
x
−20
−10
10
20
30
40
50
60
70
80
y
−20
−10
10
20
30
40
50
60
70
80
Cartesian Coordinate PlaneThe horizontal axis is labeled with x. The horizontal axis contains 21 grid lines, starts at -20, and ends at 80; and the axis is labeled with the values: -20, -10, 10, 20, 30, 40, 50, 60, 70, 80.The vertical axis is labeled with y. The vertical axis contains 21 grid lines, starts at -20, and ends at 80; and the axis is labeled with the values: -20, -10, 10, 20, 30, 40, 50, 60, 70, 80.System of EquationsThe system contains the graphs of 2 equations with 1 points of intersection.Intersection 1Line 1y-Intercept 1Line 2y-Intercept 2


To solve the system of equations graphically, we'll first rewrite each equation in slope-intercept form \((y = mx + b)\).

### Step 1: Write Equations in Slope-Intercept Form

Given the system of equations:
1. \(x + y = 75\)
2. \(x - y = 15\)

Let's rearrange each equation to isolate \(y\).

#### Equation 1: \(x + y = 75\)

Solve for \(y\):
\[
y = -x + 75
\]

- **Slope**: \(m_1 = -1\)
- **Y-Intercept**: \(b_1 = 75\)

#### Equation 2: \(x - y = 15\)

Solve for \(y\):
\[
-y = -x + 15 \implies y = x - 15
\]

- **Slope**: \(m_2 = 1\)
- **Y-Intercept**: \(b_2 = -15\)

### Step 2: Graph the Equations

Now, we can graph these two lines on the Cartesian coordinate plane.

1. **Line 1**: \(y = -x + 75\)
   - Slope: \(-1\)
   - Y-intercept: \(75\)

2. **Line 2**: \(y = x - 15\)
   - Slope: \(1\)
   - Y-intercept: \(-15\)

### Step 3: Find the Intersection Point

To solve graphically, we find the point where the two lines intersect. This point is the solution to the system of equations.

Let's set the equations equal to each other to find the intersection:

\[
-x + 75 = x - 15
\]

Solve for \(x\):

\[
75 + 15 = 2x \implies 90 = 2x \implies x = 45
\]

Substitute \(x = 45\) back into one of the equations to find \(y\):

\[
y = -x + 75 \implies y = -45 + 75 = 30
\]

### Solution

- **First Coordinate (x):** 45
- **Second Coordinate (y):** 30

Thus, the two numbers are **45** and **30**.

### Graph Interpretation

The lines intersect at the point \((45, 30)\). This is the only solution to the system, which means the system has a single solution.

Would you like me to plot this graph for you or do you have any questions?

---

**Relative Questions:**

1. How do you find the intersection point of two lines algebraically?
2. What does the slope of a line represent in a real-world context?
3. Can a system of equations have more than one solution? If so, under what conditions?
4. How can you verify the solution to a system of equations graphically?
5. What happens when the slopes of two lines are the same?

**Tip:** When solving a system of linear equations graphically, always look for the point of intersection, as it represents the values of the variables that satisfy both equations.

The sum of two numbers is 75 and their difference is 15. What are the two numbers? Let x= one number and y= the other number. The corresponding modeling system is {x+y=75, x−y=15}. Solve the system graphically.

Learn how to solve the system of linear equations x + y = 75 and x - y = 15 graphically. This guide includes step-by-step instructions to find the intersection point of the two lines, covering key algebraic concepts and formulas. Suitable for students in Grades 7-10.

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