To determine graphically whether the system of equations
[tex]\[
\begin{cases}
x - 2y = 2, \\
4x - 2y = 5
\end{cases}
\][/tex]
is consistent or inconsistent, we need to graph the two lines represented by these equations and observe their point of intersection.
1. Convert the equations to slope-intercept form:
For the first equation [tex]\( x - 2y = 2 \)[/tex]:
[tex]\[
\begin{align*}
x - 2y &= 2 \\
-2y &= -x + 2 \\
y &= \frac{1}{2}x - 1
\end{align*}
\][/tex]
For the second equation [tex]\( 4x - 2y = 5 \)[/tex]:
[tex]\[
\begin{align*}
4x - 2y &= 5 \\
-2y &= -4x + 5 \\
y &= 2x - \frac{5}{2}
\end{align*}
\][/tex]
2. Plot the lines on a coordinate plane:
- For the first equation [tex]\( y = \frac{1}{2}x - 1 \)[/tex]:
- The y-intercept is [tex]\(-1\)[/tex] (point [tex]\((0, -1)\)[/tex]).
- The slope is [tex]\(\frac{1}{2}\)[/tex], which means for every increase of 2 units in [tex]\(x\)[/tex], [tex]\(y\)[/tex] increases by 1 unit.
- For the second equation [tex]\( y = 2x - \frac{5}{2} \)[/tex]:
- The y-intercept is [tex]\(-\frac{5}{2}\)[/tex] or [tex]\(-2.5\)[/tex] (point [tex]\((0, -2.5)\)[/tex]).
- The slope is [tex]\(2\)[/tex], which means for every increase of 1 unit in [tex]\(x\)[/tex], [tex]\(y\)[/tex] increases by 2 units.
3. Identify the point of intersection:
By plotting the lines, you will see that they intersect at the point where both equations are satisfied simultaneously. The intersection point is [tex]\((1, -\frac{1}{2})\)[/tex].
4. Check consistency:
Since the lines intersect at a single point [tex]\((1, -\frac{1}{2})\)[/tex], this means the system of equations is consistent. There is exactly one solution to the system.
Hence, the system of equations [tex]\( x - 2y = 2 \)[/tex] and [tex]\( 4x - 2y = 5 \)[/tex] is "consistent," with the solution being [tex]\( x = 1 \)[/tex] and [tex]\( y = -\frac{1}{2} \)[/tex].
