Sure! Let's solve the system of linear equations step-by-step. The given equations are:
1) [tex]\( y = 2x - 1 \)[/tex]
2) [tex]\( y = 5 - x \)[/tex]
To find the solution, we need to determine the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] where both equations intersect.
### Step-by-Step Solution:
1. Setting the equations equal to each other:
Since both equations equal [tex]\(y\)[/tex], we can set them equal to each other:
[tex]\[
2x - 1 = 5 - x
\][/tex]
2. Solving for [tex]\(x\)[/tex]:
Combine like terms to solve for [tex]\(x\)[/tex]. Add [tex]\(x\)[/tex] to both sides:
[tex]\[
2x + x - 1 = 5
\][/tex]
Simplify:
[tex]\[
3x - 1 = 5
\][/tex]
Add 1 to both sides:
[tex]\[
3x = 6
\][/tex]
Finally, divide by 3:
[tex]\[
x = 2
\][/tex]
3. Finding [tex]\(y\)[/tex] using one of the original equations:
Substitute [tex]\(x = 2\)[/tex] into either of the two original equations. Let's use [tex]\(y = 2x - 1\)[/tex]:
[tex]\[
y = 2(2) - 1 = 4 - 1 = 3
\][/tex]
### Solution:
The solution for the system of equations is:
[tex]\[
(x, y) = (2, 3)
\][/tex]
This means the two lines intersect at the point [tex]\((2, 3)\)[/tex].
### Graphing the equation [tex]\(y = 2x - 1\)[/tex]:
1. Locate the y-intercept:
The y-intercept is where [tex]\(x = 0\)[/tex]. For the equation [tex]\(y = 2x - 1\)[/tex]:
[tex]\[
y = 2(0) - 1 = -1
\][/tex]
So the y-intercept is (0, -1).
2. Use the slope:
The slope of the equation [tex]\(y = 2x - 1\)[/tex] is 2. This means for every increase of 1 in [tex]\(x\)[/tex], [tex]\(y\)[/tex] increases by 2.
3. Plotting another point using the slope:
Starting from the y-intercept (0, -1), move 1 unit to the right (increase [tex]\(x\)[/tex] by 1) and 2 units up (increase [tex]\(y\)[/tex] by 2). This gives the point (1, 1).
4. Draw the line:
Connect the y-intercept (0, -1) and the point (1, 1), and extend the line in both directions.
By graphing both equations, you would visually find that they intersect at the point (2, 3). This confirms our algebraic solution.
