Sure, let's solve the system of equations step by step and plot the lines on the graph.
The system of equations is:
1. [tex]\( y = 2x - 1 \)[/tex]
2. [tex]\( x + 3y = 18 \)[/tex]
### Step 1: Plot the first equation [tex]\( y = 2x - 1 \)[/tex].
To plot this line, we need at least two points.
- When [tex]\( x = 0 \)[/tex]:
[tex]\[
y = 2(0) - 1 = -1
\][/tex]
Point: [tex]\( (0, -1) \)[/tex].
- When [tex]\( x = 3 \)[/tex]:
[tex]\[
y = 2(3) - 1 = 6 - 1 = 5
\][/tex]
Point: [tex]\( (3, 5) \)[/tex].
Now you can draw the line passing through the points [tex]\( (0, -1) \)[/tex] and [tex]\( (3, 5) \)[/tex].
### Step 2: Plot the second equation [tex]\( x + 3y = 18 \)[/tex].
Again, we need at least two points to plot this line.
- When [tex]\( x = 0 \)[/tex]:
[tex]\[
0 + 3y = 18 \implies y = 6
\][/tex]
Point: [tex]\( (0, 6) \)[/tex].
- When [tex]\( y = 0 \)[/tex]:
[tex]\[
x + 3(0) = 18 \implies x = 18
\][/tex]
Point: [tex]\( (18, 0) \)[/tex].
Now you can draw the line passing through the points [tex]\( (0, 6) \)[/tex] and [tex]\( (18, 0) \)[/tex].
### Step 3: Find the intersection point.
The intersection point of these two lines represents the solution to the system of equations. After plotting, you will see that the lines intersect at the point [tex]\( (3, 5) \)[/tex].
### Conclusion
So, the solution to the system of equations is [tex]\( x = 3 \)[/tex] and [tex]\( y = 5 \)[/tex].
To verify, you can substitute [tex]\( x = 3 \)[/tex] and [tex]\( y = 5 \)[/tex] back into the original equations:
1. [tex]\( y = 2x - 1 \)[/tex]
[tex]\[
5 = 2(3) - 1 \implies 5 = 5 \quad \text{(True!)}
\][/tex]
2. [tex]\( x + 3y = 18 \)[/tex]
[tex]\[
3 + 3(5) = 18 \implies 3 + 15 = 18 \implies 18 = 18 \quad \text{(True!)}
\][/tex]
Hence, the point [tex]\( (3, 5) \)[/tex] is the correct solution to the system.
