Question

question 6. the graph of a line on a coordinate plane has a slope of \\(\frac{1}{2}\\) and passes through the point \\((4, -3)\\). which equation describes the line? a. \\(x + 2y = 10\\) b. \\(x - 2y = 2\\) c. \\(x - 2y = 7\\) d. \\(x - 2y = 10\\)

Explanation:

Step1: Recall point - slope form

The point - slope form of a line is $y - y_1=m(x - x_1)$, where $m$ is the slope and $(x_1,y_1)$ is a point on the line. Given $m=\frac{1}{2}$ and the point $(4,-3)$, substitute into the formula:
$y - (-3)=\frac{1}{2}(x - 4)$
Simplify: $y + 3=\frac{1}{2}x-2$

Step2: Convert to standard form

Subtract $\frac{1}{2}x$ from both sides: $y-\frac{1}{2}x+ 3=-2$
Subtract 3 from both sides: $y-\frac{1}{2}x=-5$
Multiply every term by 2 to eliminate the fraction: $2y - x=-10$
Rearrange to get $x - 2y = 10$? Wait, no, let's check again. Wait, from $y + 3=\frac{1}{2}(x - 4)$, distribute the $\frac{1}{2}$: $y+3=\frac{1}{2}x - 2$. Then subtract $\frac{1}{2}x$: $-\frac{1}{2}x+y=-5$. Multiply by - 2: $x - 2y = 10$? Wait, no, let's check the options. Wait, the options are:
A. $x + 2y=10$
B. $x - 2y=2$
C. $x - 2y=7$
D. $x - 2y=10$

Wait, let's re - do the point - slope form. $y - y_1=m(x - x_1)$, $y_1=-3$, $x_1 = 4$, $m=\frac{1}{2}$. So $y-(-3)=\frac{1}{2}(x - 4)\implies y + 3=\frac{1}{2}x-2$. Then $y=\frac{1}{2}x-5$. Now, let's convert to standard form $Ax+By = C$. Multiply both sides by 2: $2y=x - 10\implies x-2y = 10$. So the equation is $x - 2y=10$, which is option D. Wait, but let's check by plugging the point $(4,-3)$ into option D: $4-2\times(-3)=4 + 6 = 10$, which matches. Let's check the slope of option D. Rewrite $x - 2y=10$ as $y=\frac{1}{2}x-5$, slope is $\frac{1}{2}$, which is correct.

Wait, maybe I made a mistake in the sign earlier. Let's re - derive:

Starting from $y=mx + b$. We know $m=\frac{1}{2}$, and the point $(4,-3)$. Substitute into $y=mx + b$: $-3=\frac{1}{2}(4)+b\implies - 3 = 2 + b\implies b=-5$. So the equation is $y=\frac{1}{2}x-5$. Now, convert to standard form: $x-2y = 10$ (multiply $y=\frac{1}{2}x - 5$ by 2: $2y=x - 10\implies x-2y = 10$). So the correct option is D.

Answer:

D. $x - 2y = 10$