To determine graphically whether the system of equations
[tex]\[x - 2y = 2\][/tex]
[tex]\[4x - 2y = 5\][/tex]
is consistent or inconsistent, we need to follow these steps:
### Step 1: Rewrite the Equations in Slope-Intercept Form
To graph these equations, it is useful to rewrite them in the slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept.
#### First Equation: [tex]\(x - 2y = 2\)[/tex]
1. Start with the equation [tex]\(x - 2y = 2\)[/tex].
2. Solve for [tex]\(y\)[/tex]:
[tex]\[
x - 2y = 2 \implies -2y = -x + 2 \implies y = \frac{1}{2}x - 1
\][/tex]
#### Second Equation: [tex]\(4x - 2y = 5\)[/tex]
1. Start with the equation [tex]\(4x - 2y = 5\)[/tex].
2. Solve for [tex]\(y\)[/tex]:
[tex]\[
4x - 2y = 5 \implies - 2y = -4x + 5 \implies y = 2x - \frac{5}{2}
\][/tex]
### Step 2: Plot the Equations on a Graph
Now, we can graph the equations [tex]\(y = \frac{1}{2}x - 1\)[/tex] and [tex]\(y = 2x - \frac{5}{2}\)[/tex].
1. Graph for [tex]\(y = \frac{1}{2}x - 1\)[/tex]
- The y-intercept is [tex]\(-1\)[/tex].
- The slope is [tex]\(\frac{1}{2}\)[/tex], meaning for each unit increase in [tex]\(x\)[/tex], [tex]\(y\)[/tex] increases by [tex]\(\frac{1}{2}\)[/tex].
- To plot the line, start at [tex]\((0, -1)\)[/tex] and use the slope to find another point. For instance, at [tex]\(x = 2\)[/tex], [tex]\(y = \frac{1}{2}(2) - 1 = 0\)[/tex].
2. Graph for [tex]\(y = 2x - \frac{5}{2}\)[/tex]
- The y-intercept is [tex]\(-\frac{5}{2}\)[/tex].
- The slope is [tex]\(2\)[/tex], meaning for each unit increase in [tex]\(x\)[/tex], [tex]\(y\)[/tex] increases by [tex]\(2\)[/tex].
- To plot the line, start at [tex]\((0, -\frac{5}{2})\)[/tex] and use the slope to find another point. For instance, at [tex]\(x = 1\)[/tex], [tex]\(y = 2(1) - \frac{5}{2} = -\frac{1}{2}\)[/tex].
### Step 3: Analyze the Graph
Plot both lines on the same set of axes:
- The first line (from [tex]\(y = \frac{1}{2}x - 1\)[/tex]) will start at [tex]\((0, -1)\)[/tex] and pass through points like [tex]\((2, 0)\)[/tex].
- The second line (from [tex]\(y = 2x - \frac{5}{2}\)[/tex]) will start at [tex]\((0, -\frac{5}{2})\)[/tex] and pass through points like [tex]\((1, -\frac{1}{2})\)[/tex].
### Step 4: Determine Consistency
Consistency: A system is consistent if the lines intersect at any single point, meaning there is at least one solution to the system.
When plotted, we observe that these two lines intersect at a specific point on the graph. This indicates that the system of equations has a unique solution, confirming that the system of equations is consistent.
