Which graph shows the solution of −3x−y=3

Question

and y=−3x+4
?(1 point)
Responses

Two intersecting lines are graphed across a coordinate plane.
Image with alt text: Two intersecting lines are graphed across a coordinate plane.

Two intersecting lines are graphed across a coordinate plane.
Image with alt text: Two intersecting lines are graphed across a coordinate plane.

Two intersecting lines are graphed across a coordinate plane.
Image with alt text: Two intersecting lines are graphed across a coordinate plane.

Two parallel lines are graphed across a coordinate plane. One line is dotted while the other is solid.
Image with alt text: Two parallel lines are graphed across a coordinate plane. One line is dotted while the other is solid.
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Answer 1

To determine what the graphs of the given equations look like, we can analyze the two equations:

1. Equation 1: \(-3x - y = 3\)

   • To express this in slope-intercept form (\(y = mx + b\)), we can rearrange it: \[ -y = 3 + 3x \implies y = -3x - 3 \]
   • This line has a slope of \(-3\) and a y-intercept at \(-3\).

2. Equation 2: \(y = -3x + 4\)

   • This is already in slope-intercept form with a slope of \(-3\) and a y-intercept at \(4\).

Now comparing the two lines:

• Both equations have the same slope of \(-3\), which means the lines are parallel.
• The y-intercept of the first line is \(-3\), while the y-intercept of the second line is \(4\). Since they have different y-intercepts, they will never intersect.

Thus, the correct description of the graph is that of two parallel lines. Since one is described as solid and the other as dotted, this reinforces that they do not intersect.

Therefore, the correct answer is: Two parallel lines are graphed across a coordinate plane. One line is dotted while the other is solid.

Answer 2

Use the image to answer the 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

an infinite number of solutions
an infinite number of solutions

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

(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

no solution
no solution
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Answer 3

To find the solution of the two equations \((72)x - (92) = 2y\) and \(4y = 7x - 9\), we first need to simplify each equation.

1. Equation 1: \((72)x - (92) = 2y\)

   • Rearranging it into slope-intercept form \(y = mx + b\): \[ 2y = (72)x - (92) \implies y = \frac{72}{2}x - \frac{92}{2} \implies y = 36x - 46 \]

2. Equation 2: \(4y = 7x - 9\)

   • Rearranging it into slope-intercept form: \[ y = \frac{7}{4}x - \frac{9}{4} \]

Now let's compare the two equations:

• The first equation is \(y = 36x - 46\).
• The second equation is \(y = \frac{7}{4}x - \frac{9}{4}\).

Since the two equations are clearly not in the same form and have different slopes (36 and \(\frac{7}{4}\)), they represent different lines.

However, if Emma and Angelle both ended up with the exact same graph, it suggests that their equations, though initially different, are equivalent. This would mean every point on one line is also on the other line.

In this case, both lines being equivalent means they share all their points in common.

Therefore, the correct response to the solution of the two equations is: an infinite number of solutions.

Answer 4

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

(3,0)
left parenthesis 3 comma 0 right parenthesis

There is no solution.
There is no solution.

(2,−1)

Answer 5

To find the solution for the system of equations from the table provided, we need to look at the x- and y-values corresponding to both equations:

1. Equation 1: \(y = 2x - 5\)
2. Equation 2: \(y = x - 3\)

From the table, the relevant x and y values for both equations give the following:

• 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 we look for where the y-values from both equations are equal (that is, where the two lines intersect):

• At \(x = 2\), both equations yield \(y = -1\).
• Therefore, the point \((2, -1)\) is a solution to the system of equations.

Let's check the other values:

• At \(x = 3\), the first equation gives \(y = 1\) and the second gives \(y = 0\), which are not equal.
• No other x-values yield equal y-values.

Thus, the solution to the system is \((2, -1)\).

Therefore, the correct answer is: (2,−1).

Answer 6

Use the image to answer the question.

A coordinate plane's x-axis ranges from negative 5 to 5 and its y-axis ranges from negative 3 to 7, both in 1-unit increments. A parabola and a line are plotted. The line and parabola intersect at two points. 1 point of intersection is in quadrant 1 and the other is in quadrant 2.

Which of the following shows the solution(s) for this system of equations?

(1 point)
Responses

(−2,4)
and (2,4)
left parenthesis negative 2 comma 4 right parenthesis and left parenthesis 2 comma 4 right parenthesis

There are no solutions.
There are no solutions.

(−3,0)
and (3,0)
left parenthesis negative 3 comma 0 right parenthesis and left parenthesis 3 comma 0 right parenthesis

(0,8)
left parenthesis 0 comma 8 right parenthesis
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Answer 7

If there is a parabola and a line intersecting at two points, it indicates that the system of equations has two solutions. Since you mentioned that one point of intersection is in quadrant 1 and the other is in quadrant 2, this suggests that the y-values at those intersection points are both positive (in quadrant 1) and that one of the x-values is negative (in quadrant 2).

Now, let's analyze the possible answers provided:

1. (−2,4) and (2,4):

   • This suggests two points, one to the left of the y-axis (in quadrant 2) and one to the right of the y-axis (in quadrant 1) at the same y-value of \(4\). This is a plausible scenario for the line and parabola intersecting at two points.

2. There are no solutions.:

   • This option is incorrect since you've stated there are two points of intersection.

3. (−3,0) and (3,0):

   • This suggests two points on the x-axis. Although this describes two solutions, they are on the x-axis (where \(y=0\)), which does not match with the earlier clues stating that one intersection is in quadrant 1 (where \(y\) is positive).

4. (0,8):

   • This only provides a single point which does not match your description of two intersection points.

Given this analysis, the best answer is the first option, as it describes two intersection points with appropriate coordinates across both quadrants.

Therefore, the correct response is: (−2,4) and (2,4).

Answer 8

Which equation describes an infinite solution system with the equation y=4x−3
?(1 point)
Responses

−2y+8x=−6
negative 2 y plus 8 x equals negative 6

2y−8x=−6
2 y minus 8 x equals negative 6

2y−8x=6
2 y minus 8 x equals 6

2y+8x=−6

Answer 9

To determine which equation describes a system that has infinitely many solutions with the equation \(y = 4x - 3\), we need to check if any of the provided equations are equivalent to the original equation.

An equivalent equation to \(y = 4x - 3\) will be a rearrangement of this line. For the systems of equations to have infinite solutions, the equations must represent the same line.

Let's start by rearranging the given equation into the standard form \(Ax + By = C\).

Original Equation:

1. Start with: \[ y = 4x - 3 \]
2. Rearranging gives: \[ -4x + y = -3 \implies 4x - y = 3 \]

Now, let’s check each provided equation to see if it is equivalent to \(4x - y = 3\):

1. Equation: \(-2y + 8x = -6\)

   • Rearranging gives: \[ -2y = -8x - 6 \implies y = 4x + 3 \]
   • This does NOT match \(y = 4x - 3\).

2. Equation: \(2y - 8x = -6\)

   • Rearranging gives: \[ 2y = 8x - 6 \implies y = 4x - 3 \]
   • This IS equivalent to the original equation.

3. Equation: \(2y - 8x = 6\)

   • Rearranging gives: \[ 2y = 8x + 6 \implies y = 4x + 3 \]
   • This does NOT match \(y = 4x - 3\).

4. Equation: \(2y + 8x = -6\)

   • Rearranging gives: \[ 2y = -8x - 6 \implies y = -4x - 3 \]
   • This does NOT match \(y = 4x - 3\).

Conclusion:

The equation that describes an infinite solution system with the equation \(y = 4x - 3\) is: \(2y - 8x = -6\).