Question
on the set of aces below solve the following system of equations graphically and steate the coordinates of all points in the soloution set
y=x^2+10x+24
6x+3y=-24
where do you put the graph
give me the right answer
y=x^2+10x+24
6x+3y=-24
where do you put the graph
give me the right answer
Answers
GPT-4o mini
To solve the system of equations graphically, we can plot the graphs of the two equations and find their points of intersection.
### Equations:
1. \( y = x^2 + 10x + 24 \)
2. \( 6x + 3y = -24 \) (we will rewrite this in the slope-intercept form to better plot it)
**Step 1: Rewrite the second equation:**
\[
6x + 3y = -24
\]
Dividing the entire equation by 3 to simplify:
\[
2x + y = -8
\]
Rearranging gives:
\[
y = -2x - 8
\]
### Step 2: Plotting the equations
**Equation 1: \( y = x^2 + 10x + 24 \)**
This is a parabola that opens upwards. You can find some key points by choosing values for \(x\):
- For \(x = -10\):
\[
y = (-10)^2 + 10(-10) + 24 = 100 - 100 + 24 = 24 \quad \Rightarrow \quad (-10, 24)
\]
- For \(x = -5\):
\[
y = (-5)^2 + 10(-5) + 24 = 25 - 50 + 24 = -1 \quad \Rightarrow \quad (-5, -1)
\]
- For \(x = 0\):
\[
y = (0)^2 + 10(0) + 24 = 24 \quad \Rightarrow \quad (0, 24)
\]
- For \(x = -4\):
\[
y = (-4)^2 + 10(-4) + 24 = 16 - 40 + 24 = 0 \quad \Rightarrow \quad (-4, 0)
\]
**Equation 2: \( y = -2x - 8 \)**
This is a straight line. For this line, find the intercepts and a couple more points:
- For \(x = 0\):
\[
y = -2(0) - 8 = -8 \quad \Rightarrow \quad (0, -8)
\]
- For \(x = -2\):
\[
y = -2(-2) - 8 = 4 - 8 = -4 \quad \Rightarrow \quad (-2, -4)
\]
- For \(x = -4\):
\[
y = -2(-4) - 8 = 8 - 8 = 0 \quad \Rightarrow \quad (-4, 0)
\]
Now we have several points to graph both equations.
### Step 3: Finding the points of intersection
You can calculate the intersection points either graphically or algebraically. Setting the equations equal:
\[
x^2 + 10x + 24 = -2x - 8
\]
Rearranging gives:
\[
x^2 + 12x + 32 = 0
\]
Now we factor it:
\[
(x + 4)(x + 8) = 0
\]
So, \( x = -4 \) or \( x = -8 \).
1. **For \( x = -4 \)**:
\[
y = -2(-4) - 8 = 8 - 8 = 0 \quad \Rightarrow \quad (-4, 0)
\]
2. **For \( x = -8 \)**:
\[
y = -2(-8) - 8 = 16 - 8 = 8 \quad \Rightarrow \quad (-8, 8)
\]
### Points of Intersection
The solution set contains the following points:
1. \( (-4, 0) \)
2. \( (-8, 8) \)
### Conclusion
To summarize:
- The coordinates of all points in the solution set are **(-4, 0)** and **(-8, 8)**.
- You can graph these equations on graphing software or graph paper, marking the points and the curves, and highlighting where the two equations intersect.
### Equations:
1. \( y = x^2 + 10x + 24 \)
2. \( 6x + 3y = -24 \) (we will rewrite this in the slope-intercept form to better plot it)
**Step 1: Rewrite the second equation:**
\[
6x + 3y = -24
\]
Dividing the entire equation by 3 to simplify:
\[
2x + y = -8
\]
Rearranging gives:
\[
y = -2x - 8
\]
### Step 2: Plotting the equations
**Equation 1: \( y = x^2 + 10x + 24 \)**
This is a parabola that opens upwards. You can find some key points by choosing values for \(x\):
- For \(x = -10\):
\[
y = (-10)^2 + 10(-10) + 24 = 100 - 100 + 24 = 24 \quad \Rightarrow \quad (-10, 24)
\]
- For \(x = -5\):
\[
y = (-5)^2 + 10(-5) + 24 = 25 - 50 + 24 = -1 \quad \Rightarrow \quad (-5, -1)
\]
- For \(x = 0\):
\[
y = (0)^2 + 10(0) + 24 = 24 \quad \Rightarrow \quad (0, 24)
\]
- For \(x = -4\):
\[
y = (-4)^2 + 10(-4) + 24 = 16 - 40 + 24 = 0 \quad \Rightarrow \quad (-4, 0)
\]
**Equation 2: \( y = -2x - 8 \)**
This is a straight line. For this line, find the intercepts and a couple more points:
- For \(x = 0\):
\[
y = -2(0) - 8 = -8 \quad \Rightarrow \quad (0, -8)
\]
- For \(x = -2\):
\[
y = -2(-2) - 8 = 4 - 8 = -4 \quad \Rightarrow \quad (-2, -4)
\]
- For \(x = -4\):
\[
y = -2(-4) - 8 = 8 - 8 = 0 \quad \Rightarrow \quad (-4, 0)
\]
Now we have several points to graph both equations.
### Step 3: Finding the points of intersection
You can calculate the intersection points either graphically or algebraically. Setting the equations equal:
\[
x^2 + 10x + 24 = -2x - 8
\]
Rearranging gives:
\[
x^2 + 12x + 32 = 0
\]
Now we factor it:
\[
(x + 4)(x + 8) = 0
\]
So, \( x = -4 \) or \( x = -8 \).
1. **For \( x = -4 \)**:
\[
y = -2(-4) - 8 = 8 - 8 = 0 \quad \Rightarrow \quad (-4, 0)
\]
2. **For \( x = -8 \)**:
\[
y = -2(-8) - 8 = 16 - 8 = 8 \quad \Rightarrow \quad (-8, 8)
\]
### Points of Intersection
The solution set contains the following points:
1. \( (-4, 0) \)
2. \( (-8, 8) \)
### Conclusion
To summarize:
- The coordinates of all points in the solution set are **(-4, 0)** and **(-8, 8)**.
- You can graph these equations on graphing software or graph paper, marking the points and the curves, and highlighting where the two equations intersect.