Use the image to answer the question.

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

(1.286,0)
left parenthesis 1.286 comma 0 right parenthesis

no solution
no solution

an infinite number of solutions
an infinite number of solutions

(1.286,0) and (0,−2.25)

GPT-4o mini · Answer

To find the solution of the equations $72x - 92 = 2y$ and $4y = 7x - 9$, we need to first rewrite both equations in a more manageable form. 1. **Rearranging the first equation**: $$ 72x - 92 = 2y $$ We can solve for $y$: $$ 2y = 72x - 92 \quad \Rightarrow \quad y = 36x - 46 $$ 2. **Rearranging the second equation**: $$ 4y = 7x - 9 $$ Solving for $y$: $$ y = \frac{7}{4}x - \frac{9}{4} $$ Now we can see the slopes and y-intercepts of both lines: - The first equation $y = 36x - 46$ has a slope of 36 and a y-intercept of -46. - The second equation $y = \frac{7}{4}x - \frac{9}{4}$ has a slope of $1.75$ and a y-intercept of $-2.25$. To determine if the lines have the same graph (infinite solutions) or if they intersect at a specific point (single solution), we need to find points of intersection. Given that Emma and Angelle end up with the same graph, their equations must represent the same line. Let's solve for $x$ and $y$: We set both $y$ equations equal to each other: $$ 36x - 46 = \frac{7}{4}x - \frac{9}{4} $$ To eliminate the fraction, multiply the whole equation by 4: $$ 4(36x - 46) = 7x - 9 $$ $$ 144x - 184 = 7x - 9 $$ $$ 144x - 7x = 184 - 9 $$ $$ 137x = 175 \quad \Rightarrow \quad x = \frac{175}{137} \approx 1.286 $$ Now we can substitute this value of $x$ back into either equation to find $y$. We'll use the first equation: $$ y = 36\left(\frac{175}{137}\right) - 46 $$ Calculating $y$: $$ y = \frac{6300}{137} - \frac{6302}{137} = \frac{-2}{137} \approx 0 $$ Thus, the solution of the system of equations is approximately $(1.286, 0)$. Since the lines overlap completely, the answer is: **an infinite number of solutions** (Since they are the same line). However, if specific solutions are asked, we can provide the calculated point: $(1.286, 0)$.

adaleta · Answer

Use the table to answer the question.

x-value y-value if y=2x−5 y-value if y=x−3
0 −5 −3
1 −3 −2
2 −1 −1
3 1 0
Jian made a table showing some x- and y-values in a system of equations. Use his table to identify the solution for the system.

(1 point)
Responses

There is no solution.
There is no solution.

There is an infinite number of solutions.
There is an infinite number of solutions.

(2,−1)
left parenthesis 2 comma negative 1 right parenthesis

(3,0)

GPT-4o mini · Answer

To identify the solution for the system of equations from Jian's table, we need to check the provided y-values for each corresponding x-value from the equations $y = 2x - 5$ and $y = x - 3$.

Let's review the x-values and the corresponding y-values:

For $x = 0$:
- $y = 2(0) - 5 = -5$
- $y = 0 - 3 = -3$
For $x = 1$:
- $y = 2(1) - 5 = -3$
- $y = 1 - 3 = -2$
For $x = 2$:
- $y = 2(2) - 5 = -1$
- $y = 2 - 3 = -1$
For $x = 3$:
- $y = 2(3) - 5 = 1$
- $y = 3 - 3 = 0$

Now, let's analyze the y-values for common x-values:

At $x = 2$, both equations yield the same y-value:
- $y = -1$ (from both equations)

This means that the point $(2, -1)$ is a solution to the system of equations.

Next, we need to check if there are any other common solutions:

For $x = 0$ and $x = 1$, the y-values from both equations do not match.
For $x = 3$, the y-values again do not match ($y = 1$ for the first equation and $y = 0$ for the second).

Since $(2, -1)$ is the only point where the y-values for both equations match, this point is the only solution to the system.

Therefore, the correct response is:

(2, -1).