Question
One line, graphed as both solid and a dotted, is graphed on a coordinate plane. The x-axis ranges from negative 10 to 10 in increments of 1. The y-axis ranges from negative 10 to 10 in increments of 1.
In math class, Emma and Angelle pair as a group. Emma’s task is to graph (72)x−(92)=2y
and Angelle’s task is to graph 4y=7x−9
. After they complete their graphs, they compare their work. They both get exactly the same graph. What is the solution of (72)x−(92)=2y
and 4y=7x−9
?
(1 point)
Responses
no solution
no solution
(1.286,0)
left parenthesis 1.286 comma 0 right parenthesis
an infinite number of solutions
an infinite number of solutions
(1.286,0)
and (0,−2.25)
left parenthesis 1.286 comma 0 right parenthesis and left parenthesis 0 comma negative 2.25 right parenthesis
In math class, Emma and Angelle pair as a group. Emma’s task is to graph (72)x−(92)=2y
and Angelle’s task is to graph 4y=7x−9
. After they complete their graphs, they compare their work. They both get exactly the same graph. What is the solution of (72)x−(92)=2y
and 4y=7x−9
?
(1 point)
Responses
no solution
no solution
(1.286,0)
left parenthesis 1.286 comma 0 right parenthesis
an infinite number of solutions
an infinite number of solutions
(1.286,0)
and (0,−2.25)
left parenthesis 1.286 comma 0 right parenthesis and left parenthesis 0 comma negative 2.25 right parenthesis
Answers
GPT-4o mini
To determine the solution of the equations \( (72)x - (92) = 2y \) and \( 4y = 7x - 9 \), we can start by rearranging both equations into slope-intercept form (y = mx + b) and see if they are equivalent.
Starting with the first equation:
1. Rearrange \( (72)x - (92) = 2y \):
\[
2y = 72x - 92
\]
\[
y = 36x - 46
\]
Next, for the second equation:
2. Rearrange \( 4y = 7x - 9 \):
\[
y = \frac{7}{4}x - \frac{9}{4}
\]
Now we check if both equations represent the same line. We observe their slopes (36 and 1.75) and y-intercepts (-46 and -2.25). Since these values are not equal, the lines represented by these equations are not the same.
Since they intersect at a specific point, we can find that point by setting the equations equal to each other.
Set the y-values equal to each other (as we have both in y = mx + b form):
\[
36x - 46 = \frac{7}{4}x - \frac{9}{4}
\]
To eliminate the fraction, multiply through by 4:
\[
4(36x - 46) = 7x - 9
\]
\[
144x - 184 = 7x - 9
\]
Now, combine like terms:
\[
144x - 7x = -9 + 184
\]
\[
137x = 175
\]
\[
x = \frac{175}{137} \approx 1.28
\]
Now substitute \( x \) back into either equation to find \( y \):
Using \( y = 36x - 46 \):
\[
y = 36\left(\frac{175}{137}\right) - 46
\]
Calculating it yields \( y \approx 0 \).
Thus, the intersection point is roughly \( (1.286, 0) \).
Therefore, since both equations yield the same line when reduced, the final answer is that they intersect at one specific point:
**(1.286, 0)**
Starting with the first equation:
1. Rearrange \( (72)x - (92) = 2y \):
\[
2y = 72x - 92
\]
\[
y = 36x - 46
\]
Next, for the second equation:
2. Rearrange \( 4y = 7x - 9 \):
\[
y = \frac{7}{4}x - \frac{9}{4}
\]
Now we check if both equations represent the same line. We observe their slopes (36 and 1.75) and y-intercepts (-46 and -2.25). Since these values are not equal, the lines represented by these equations are not the same.
Since they intersect at a specific point, we can find that point by setting the equations equal to each other.
Set the y-values equal to each other (as we have both in y = mx + b form):
\[
36x - 46 = \frac{7}{4}x - \frac{9}{4}
\]
To eliminate the fraction, multiply through by 4:
\[
4(36x - 46) = 7x - 9
\]
\[
144x - 184 = 7x - 9
\]
Now, combine like terms:
\[
144x - 7x = -9 + 184
\]
\[
137x = 175
\]
\[
x = \frac{175}{137} \approx 1.28
\]
Now substitute \( x \) back into either equation to find \( y \):
Using \( y = 36x - 46 \):
\[
y = 36\left(\frac{175}{137}\right) - 46
\]
Calculating it yields \( y \approx 0 \).
Thus, the intersection point is roughly \( (1.286, 0) \).
Therefore, since both equations yield the same line when reduced, the final answer is that they intersect at one specific point:
**(1.286, 0)**