To rewrite the given equation [tex]\(3x + 2y = 18\)[/tex] in the form [tex]\(y = mx + c\)[/tex], we need to solve for [tex]\(y\)[/tex]. Here are the steps:
1. Start with the given equation:
[tex]\[
3x + 2y = 18
\][/tex]
2. Isolate the term involving [tex]\(y\)[/tex] by subtracting [tex]\(3x\)[/tex] from both sides:
[tex]\[
2y = 18 - 3x
\][/tex]
3. Solve for [tex]\(y\)[/tex] by dividing every term by 2:
[tex]\[
y = \frac{18 - 3x}{2}
\][/tex]
4. Simplify the equation:
[tex]\[
y = -\frac{3}{2}x + 9
\][/tex]
Now, the equation is in the form [tex]\(y = mx + c\)[/tex] where [tex]\(m\)[/tex] is the slope and [tex]\(c\)[/tex] is the y-intercept. From this equation, we can identify:
- The slope [tex]\(m = -\frac{3}{2}\)[/tex]
- The y-intercept [tex]\(c = 9\)[/tex]
Next, we need to determine if the point [tex]\((4, 3)\)[/tex] lies on the line. To do this, we substitute [tex]\(x = 4\)[/tex] into the equation and check if the resulting [tex]\(y\)[/tex]-value is 3:
1. Substitute [tex]\(x = 4\)[/tex] into the equation [tex]\(y = -\frac{3}{2}x + 9\)[/tex]:
[tex]\[
y = -\frac{3}{2} \cdot 4 + 9
\][/tex]
2. Perform the multiplication and addition:
[tex]\[
y = -6 + 9
\][/tex]
[tex]\[
y = 3
\][/tex]
Since substituting [tex]\(x = 4\)[/tex] gives us [tex]\(y = 3\)[/tex], we see that [tex]\((4, 3)\)[/tex] satisfies the equation [tex]\(y = -\frac{3}{2}x + 9\)[/tex].
Therefore, the point [tex]\((4, 3)\)[/tex] lies on the line represented by the equation. The values of [tex]\(m\)[/tex] and [tex]\(c\)[/tex] are [tex]\(-\frac{3}{2}\)[/tex] and 9, respectively, and [tex]\((4, 3)\)[/tex] is indeed on the line.
