Question
Two linear equations are represented by using the tables below.
A 2-column table with 4 rows is titled Equation A. Column 1 is labeled x with entries negative 5, negative 2, 0, 1. Column 2 is labeled y with entries negative 4, negative 1, 1, 2.
A 2-column table with 4 rows is titled Equation B. Column 1 is labeled x with entries negative 6, negative 3, 3, 6. Column 2 is labeled y with entries negative 4, negative 2, 2, 4.
The data points for equation A are plotted on the coordinate plane below and are connected by using a straight line.
On a coordinate plane, a line goes through points (negative 5, negative 4), (negative 2, negative 1), (0, 1), (1, 2).
What is the solution to the system of equations?
(–6, –4)
(–5, –4)
(–3, –2)
(0, 1)
A 2-column table with 4 rows is titled Equation A. Column 1 is labeled x with entries negative 5, negative 2, 0, 1. Column 2 is labeled y with entries negative 4, negative 1, 1, 2.
A 2-column table with 4 rows is titled Equation B. Column 1 is labeled x with entries negative 6, negative 3, 3, 6. Column 2 is labeled y with entries negative 4, negative 2, 2, 4.
The data points for equation A are plotted on the coordinate plane below and are connected by using a straight line.
On a coordinate plane, a line goes through points (negative 5, negative 4), (negative 2, negative 1), (0, 1), (1, 2).
What is the solution to the system of equations?
(–6, –4)
(–5, –4)
(–3, –2)
(0, 1)
Answers
Answered by
GPT-4o mini
To find the solution to the system of equations represented by the two tables, you need to determine the point where the two lines intersect.
1. **Determine the equations of both lines based on the given points.**
For **Equation A**, the given points are:
- (-5, -4)
- (-2, -1)
- (0, 1)
- (1, 2)
To find the equation of the line, we can use any two points to find the slope (m) and then use the point-slope form to find the equation.
Let's calculate the slope between points (-5, -4) and (-2, -1):
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1 - (-4)}{-2 - (-5)} = \frac{3}{3} = 1
\]
Using point-slope form \(y - y_1 = m(x - x_1)\):
\[
y + 4 = 1(x + 5) \implies y = x + 1
\]
So, the equation for Equation A is:
\[
y = x + 1
\]
For **Equation B**, the given points are:
- (-6, -4)
- (-3, -2)
- (3, 2)
- (6, 4)
Calculate the slope between points (-6, -4) and (-3, -2):
\[
m = \frac{-2 - (-4)}{-3 - (-6)} = \frac{2}{3} = \frac{2}{3}
\]
Using point-slope form \(y - y_1 = m(x - x_1)\):
\[
y + 4 = \frac{2}{3}(x + 6) \implies y = \frac{2}{3}x + 2 - 4 \implies y = \frac{2}{3}x - 2
\]
So, the equation for Equation B is:
\[
y = \frac{2}{3}x - 2
\]
2. **Set the equations equal to each other to find the intersection.**
\[
x + 1 = \frac{2}{3}x - 2
\]
Multiply everything by 3 to eliminate the fraction:
\[
3x + 3 = 2x - 6
\]
Rearranging gives:
\[
3x - 2x = -6 - 3 \implies x = -9
\]
3. **Substituting back to find y:**
\[
y = -9 + 1 = -8 \quad \text{if using Equation A}
\]
or
\[
y = \frac{2}{3}(-9) - 2 = -6 - 2 = -8 \quad \text{if using Equation B}
\]
4. **Conclusion:**
After solving, we found the point of intersection to be (-9, -8). However, since you are asked for the solutions mentioned in the options, we need to check for when both equations give the same value at given points.
From the given options:
- (–6, –4): Substitute in both equations: A gives -5, B gives -4; no solution.
- (–5, –4): Substitute in both: A gives -4, B gives none; no solution.
- (–3, –2): Substitute in both: A gives -2, B gives -2; solution but point not valid for lines.
- (0, 1): Substitute: A gives 1, B gives -2; no solution.
**Final answer**: Check for points that lie on both equations.
The only solution possible from the options is (–3, –2) based on equation slopes but is not intersection either. Thus, no suitable solution found in given options.
1. **Determine the equations of both lines based on the given points.**
For **Equation A**, the given points are:
- (-5, -4)
- (-2, -1)
- (0, 1)
- (1, 2)
To find the equation of the line, we can use any two points to find the slope (m) and then use the point-slope form to find the equation.
Let's calculate the slope between points (-5, -4) and (-2, -1):
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1 - (-4)}{-2 - (-5)} = \frac{3}{3} = 1
\]
Using point-slope form \(y - y_1 = m(x - x_1)\):
\[
y + 4 = 1(x + 5) \implies y = x + 1
\]
So, the equation for Equation A is:
\[
y = x + 1
\]
For **Equation B**, the given points are:
- (-6, -4)
- (-3, -2)
- (3, 2)
- (6, 4)
Calculate the slope between points (-6, -4) and (-3, -2):
\[
m = \frac{-2 - (-4)}{-3 - (-6)} = \frac{2}{3} = \frac{2}{3}
\]
Using point-slope form \(y - y_1 = m(x - x_1)\):
\[
y + 4 = \frac{2}{3}(x + 6) \implies y = \frac{2}{3}x + 2 - 4 \implies y = \frac{2}{3}x - 2
\]
So, the equation for Equation B is:
\[
y = \frac{2}{3}x - 2
\]
2. **Set the equations equal to each other to find the intersection.**
\[
x + 1 = \frac{2}{3}x - 2
\]
Multiply everything by 3 to eliminate the fraction:
\[
3x + 3 = 2x - 6
\]
Rearranging gives:
\[
3x - 2x = -6 - 3 \implies x = -9
\]
3. **Substituting back to find y:**
\[
y = -9 + 1 = -8 \quad \text{if using Equation A}
\]
or
\[
y = \frac{2}{3}(-9) - 2 = -6 - 2 = -8 \quad \text{if using Equation B}
\]
4. **Conclusion:**
After solving, we found the point of intersection to be (-9, -8). However, since you are asked for the solutions mentioned in the options, we need to check for when both equations give the same value at given points.
From the given options:
- (–6, –4): Substitute in both equations: A gives -5, B gives -4; no solution.
- (–5, –4): Substitute in both: A gives -4, B gives none; no solution.
- (–3, –2): Substitute in both: A gives -2, B gives -2; solution but point not valid for lines.
- (0, 1): Substitute: A gives 1, B gives -2; no solution.
**Final answer**: Check for points that lie on both equations.
The only solution possible from the options is (–3, –2) based on equation slopes but is not intersection either. Thus, no suitable solution found in given options.