To solve the system of equations graphically:
[tex]\[
\begin{cases}
2x - y = 5 \\
x - y = 1
\end{cases}
\][/tex]
we will follow these steps:
1. Rewrite each equation in slope-intercept form (y = mx + b) to make it easier to graph.
2. For the first equation [tex]\(2x - y = 5\)[/tex]:
- Start by isolating y.
[tex]\[
2x - y = 5
\][/tex]
[tex]\[
-y = -2x + 5
\][/tex]
[tex]\[
y = 2x - 5
\][/tex]
3. For the second equation [tex]\(x - y = 1\)[/tex]:
- Again, isolate y.
[tex]\[
x - y = 1
\][/tex]
[tex]\[
-y = -x + 1
\][/tex]
[tex]\[
y = x - 1
\][/tex]
4. Now, plot these equations on a coordinate plane:
- For [tex]\( y = 2x - 5 \)[/tex]:
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 2(0) - 5 = -5 \)[/tex]. So, one point is [tex]\( (0, -5) \)[/tex].
- When [tex]\( x = 3 \)[/tex], [tex]\( y = 2(3) - 5 = 6 - 5 = 1 \)[/tex]. So, another point is [tex]\( (3, 1) \)[/tex].
- Draw a straight line passing through these points to represent [tex]\( y = 2x - 5 \)[/tex].
- For [tex]\( y = x - 1 \)[/tex]:
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 0 - 1 = -1 \)[/tex]. So, one point is [tex]\( (0, -1) \)[/tex].
- When [tex]\( x = 3 \)[/tex], [tex]\( y = 3 - 1 = 2 \)[/tex]. So, another point is [tex]\( (3, 2) \)[/tex].
- Draw a straight line passing through these points to represent [tex]\( y = x - 1 \)[/tex].
5. Find the intersection of the two lines:
- The two lines intersect at the point where both equations are satisfied.
Upon graphing, you will see that the lines intersect at the point [tex]\((4, 3)\)[/tex].
Therefore, the solution to the system of equations is:
[tex]\[
x = 4, \quad y = 3
\][/tex]
