To graph the system of equations and determine the solution, follow these steps:
### 1. Understand the Equations
We have the following system of linear equations:
[tex]\[
\begin{cases}
y = 2x - 1 \\
y = -x + 5
\end{cases}
\][/tex]
### 2. Plot Each Equation on a Graph
#### Equation 1: [tex]\( y = 2x - 1 \)[/tex]
- Slope (m): 2
- Y-intercept (b): -1
To plot this equation, start at the y-intercept [tex]\((0, -1)\)[/tex] and use the slope to find another point. Since the slope is 2, it means for every 1 unit increase in [tex]\(x\)[/tex], [tex]\(y\)[/tex] increases by 2 units.
- Starting point: [tex]\((0, -1)\)[/tex]
- Another point: [tex]\((1, 2(1) - 1) = (1, 1)\)[/tex]
Draw a line through these points.
#### Equation 2: [tex]\( y = -x + 5 \)[/tex]
- Slope (m): -1
- Y-intercept (b): 5
To plot this equation, start at the y-intercept [tex]\((0, 5)\)[/tex] and use the slope to find another point. Since the slope is -1, it means for every 1 unit increase in [tex]\(x\)[/tex], [tex]\(y\)[/tex] decreases by 1 unit.
- Starting point: [tex]\((0, 5)\)[/tex]
- Another point: [tex]\((1, -(1) + 5) = (1, 4)\)[/tex]
Draw a line through these points.
### 3. Find the Intersection Point
To find the solution to the system, we need to determine the point where the two lines intersect. This intersection point is the solution to the system of equations.
We can solve the equations algebraically by setting them equal to each other:
[tex]\[
2x - 1 = -x + 5
\][/tex]
Add [tex]\(x\)[/tex] to both sides:
[tex]\[
2x + x - 1 = 5
\][/tex]
[tex]\[
3x - 1 = 5
\][/tex]
Add 1 to both sides:
[tex]\[
3x = 6
\][/tex]
Divide by 3:
[tex]\[
x = 2
\][/tex]
Substitute [tex]\(x = 2\)[/tex] back into either original equation to find [tex]\(y\)[/tex]:
[tex]\[
y = 2(2) - 1
\][/tex]
[tex]\[
y = 4 - 1
\][/tex]
[tex]\[
y = 3
\][/tex]
So the intersection point is [tex]\((2, 3)\)[/tex].
### 4. Conclusion
The graph you drew should show the lines intersecting at the point [tex]\((2, 3)\)[/tex]. This is the solution to the system of equations.
Thus, the answer is:
[tex]\[
\boxed{(2, 3)}
\][/tex]
