Sure, I'll guide you through the steps required to solve each part of the question using the equation [tex]\(4x + 3y = -12\)[/tex].
### Part (a): Graph the equation on a coordinate grid
1. Rewrite the Equation:
It's often helpful to rewrite the equation in slope-intercept form [tex]\(y = mx + b\)[/tex], but we'll keep it in its given form for plotting:
[tex]\[4x + 3y = -12\][/tex]
2. Find Intercepts:
- x-intercept (where [tex]\(y = 0\)[/tex]):
[tex]\[4x + 3(0) = -12 \Rightarrow 4x = -12 \Rightarrow x = -3\][/tex]
So, the x-intercept is [tex]\((-3, 0)\)[/tex].
- y-intercept (where [tex]\(x = 0\)[/tex]):
[tex]\[4(0) + 3y = -12 \Rightarrow 3y = -12 \Rightarrow y = -4\][/tex]
So, the y-intercept is [tex]\((0, -4)\)[/tex].
3. Plot Points and Draw Line:
- Plot the intercepts [tex]\((-3, 0)\)[/tex] and [tex]\((0, -4)\)[/tex] on the grid.
- Draw a straight line through these points to represent the equation [tex]\(4x + 3y = -12\)[/tex].
### Part (b): Give Three Solutions to the Equation
1. Point (0, -4):
- From the intercepts calculated above, we know that [tex]\((0, -4)\)[/tex] is a solution.
2. Point (3, -8):
- Substitute [tex]\(x = 3\)[/tex] into the equation:
[tex]\[4(3) + 3y = -12\][/tex]
[tex]\[12 + 3y = -12\][/tex]
[tex]\[3y = -24\][/tex]
[tex]\[y = -8\][/tex]
- So, [tex]\((3, -8)\)[/tex] is a solution.
3. Point (-3, 0):
- From the intercepts calculated above, we know that [tex]\((-3, 0)\)[/tex] is also a solution.
### Summary
- Three solutions for the equation [tex]\(4x + 3y = -12\)[/tex] are:
- [tex]\((0, -4)\)[/tex]
- [tex]\((3, -8)\)[/tex]
- [tex]\((-3, 0)\)[/tex]
These points all lie on the line represented by the equation, and you should graph the line by plotting these points and drawing a line through them.
