Question

Solve the following system of equations graphically on the set of axes below.
y, equals, one half, x, minus, 5
y=
2
1
​
x−5
x, plus, y, equals, 1
x+y=1

Answer 1

x+y=1

Answer 2

To solve the system of equations graphically, we will plot both equations on the same set of axes.

### 1. First Equation:
\( y = \frac{1}{2}x - 5 \)

This is in slope-intercept form \( y = mx + b \), where \( m = \frac{1}{2} \) and \( b = -5 \).

- **Y-Intercept**: The point where the line crosses the y-axis (when \( x = 0 \)):
\[
y = \frac{1}{2}(0) - 5 = -5
\]
So the y-intercept is (0, -5).

- **X-Intercept**: The point where the line crosses the x-axis (when \( y = 0 \)):
\[
0 = \frac{1}{2}x - 5 \implies \frac{1}{2}x = 5 \implies x = 10
\]
So the x-intercept is (10, 0).

### 2. Second Equation:
\( x + y = 1 \)

We can rearrange this equation to slope-intercept form \( y = mx + b \):
\[
y = -x + 1
\]

- **Y-Intercept**: When \( x = 0 \):
\[
y = -0 + 1 = 1
\]
So the y-intercept is (0, 1).

- **X-Intercept**: When \( y = 0 \):
\[
0 = -x + 1 \implies x = 1
\]
So the x-intercept is (1, 0).

### Plotting the Lines
Now, let's plot the intercepts for each equation:

1. The line for \( y = \frac{1}{2}x - 5 \) will pass through:
- (0, -5)
- (10, 0)

2. The line for \( y = -x + 1 \) will pass through:
- (0, 1)
- (1, 0)

### Finding the Point of Intersection
The solution to the system of equations is where the two lines intersect.

To find this point algebraically, set the right-hand sides of the equations equal to each other:

\[
\frac{1}{2}x - 5 = -x + 1
\]

Multiplying through by 2 to eliminate the fraction:

\[
x - 10 = -2x + 2
\]

Combine like terms:

\[
3x = 12 \implies x = 4
\]

Substituting \( x = 4 \) back into one of the original equations to find \( y \):

Using \( y = \frac{1}{2}x - 5 \):

\[
y = \frac{1}{2}(4) - 5 = 2 - 5 = -3
\]

### Intersection Point
So, the solution to the system of equations is:
\[
(4, -3)
\]

You can then plot both equations on the axes, and the point (4, -3) will be where they intersect.

Answer 3

Let's clarify and analyze the given equations again, specifically focusing on the one you just mentioned, which is part of the system.

### Given Equations
1. \( y = \frac{1}{2}x - 5 \)
2. \( x + y = 1 \) (or rearranging gives \( y = -x + 1 \))

We will solve this system graphically by finding the lines corresponding to each equation and determining the intersection.

### Step 1: Graph the First Equation
**For \( y = \frac{1}{2}x - 5 \):**
- **Y-Intercept**: When \( x = 0 \):
\[
y = \frac{1}{2}(0) - 5 = -5
\]
So the point is (0, -5).

- **X-Intercept**: When \( y = 0 \):
\[
0 = \frac{1}{2}x - 5 \implies \frac{1}{2}x = 5 \implies x = 10
\]
So the point is (10, 0).

### Step 2: Graph the Second Equation
**For \( x + y = 1 \), or rearranged as \( y = -x + 1 \):**
- **Y-Intercept**: When \( x = 0 \):
\[
y = -0 + 1 = 1
\]
So the point is (0, 1).

- **X-Intercept**: When \( y = 0 \):
\[
0 = -x + 1 \implies x = 1
\]
So the point is (1, 0).

### Step 3: Plotting the Lines
- The first line \( y = \frac{1}{2}x - 5 \) will pass through:
- (0, -5)
- (10, 0)

- The second line \( y = -x + 1 \) will pass through:
- (0, 1)
- (1, 0)

### Step 4: Finding the Intersection Point
To find the intersection algebraically, set the right sides equal:
\[
\frac{1}{2}x - 5 = -x + 1
\]

Multiply through by 2 to eliminate the fraction:
\[
x - 10 = -2x + 2
\]

Combine like terms:
\[
3x = 12 \implies x = 4
\]

Now substitute \( x = 4 \) back into either equation to find \( y \):
Using \( y = -x + 1 \):
\[
y = -4 + 1 = -3
\]

### The Intersection Point
The solution to the system of equations is:
\[
(4, -3)
\]

### Summary
Graphically, the lines will intersect at the point (4, -3). If you plot both lines on a graph, this intersection point will be where both equations are satisfied simultaneously.