Complete the statement with an equation that makes the statement true.(1 point)

Question

The system of equations 3x+4y=1
and y=
has an infinite number of solutions.....

Answer 1

For the system of equations \(3x + 4y = 1\) to have an infinite number of solutions, the second equation must be a multiple of the first.

To express \(y\) in terms of \(x\) from the equation \(3x + 4y = 1\), we can rearrange the equation:

1. Subtract \(3x\) from both sides: \[ 4y = 1 - 3x \]

2. Divide both sides by 4: \[ y = \frac{1}{4} - \frac{3}{4}x \]

To create another equation that is equivalent and thus yields infinite solutions, we can multiply the entire equation by a non-zero constant. For example, if we multiply by 2, we get:

\[ y = \frac{1}{2} - \frac{3}{2}x \]

This equation is a valid representation, but to maintain equality, we must ensure it's derived correctly from the original equation.

Thus, the statement can be completed as:

The system of equations \(3x + 4y = 1\) and \(y = -\frac{3}{4}x + \frac{1}{4}\) has an infinite number of solutions.

Alternatively, the equation \(y = k(-\frac{3}{4}x + \frac{1}{4})\) for any non-zero constant \(k\) will also work, since it represents a scale of the original linear equation.

Answer 2

Cruz wrote a system of equations with an infinite number of solutions. One of his equations is 6x+3y=9
. What could be his second equation? (Write the equation in slope-intercept form.)(1 point)
The second equation could be y=
.

Answer 3

To find a second equation that has an infinite number of solutions along with the equation \(6x + 3y = 9\), we need to make sure that the second equation is a multiple of the first.

First, let's rewrite the given equation in slope-intercept form (\(y = mx + b\)):

1. Start with the equation: \[ 6x + 3y = 9 \]

2. Subtract \(6x\) from both sides: \[ 3y = -6x + 9 \]

3. Divide every term by 3 to isolate \(y\): \[ y = -2x + 3 \]

Now, to find a second equation that has the same slope and y-intercept (which will make it identical or a multiple), we can multiply the entire equation by a non-zero constant. For example, if we multiply by 2, we will get:

\[ y = -2 \cdot 2x + 2 \cdot 3 \] \[ y = -4x + 6 \]

This is a valid second equation. Thus, Cruz's second equation could be:

The second equation could be \(y = -2x + 3\) (stating it in slope-intercept form). Alternatively, it could also be \(y = -4x + 6\) or \(y = -k(2x - 3)\) for any non-zero constant \(k\) that scales the equation.

For the simplest response to your question, we will write:

The second equation could be \(y = -2x + 3\).

Answer 4

Use the image to answer the question.

A coordinate graph goes from negative 10 to 10 in 1-unit increments on both the x- and y-axes. Two linear equations are graphed, along with two points.

The image shows the graph of the system of linear equations {y=2x+2y=2x−4
. Which of the following points, if any, are solutions to the system of equations?

(1 point)
Responses

(0,2)
open paren 0 comma 2 close paren

Neither of these points are solutions.
Neither of these points are solutions.

(0,−4)
open paren 0 comma negative 4 close paren

Both of these points are solutions.

Use the image to answer the question.



Two intersecting lines are graphed across a coordinate plane.

The graph models the swimming paths of two students, where x represents the time after a buzzer sounds and y represents height from the pool edge. One student dived into the pool from the edge of the pool 3 minutes before the buzzer went off. The second student dived into the pool from a board 4 feet above the ground right when the buzzer went off. Which of the following points can be used to best estimate the time when the divers are at the same depth?

(1 point)
Responses

(3.67, −7)
left parenthesis 3.67 comma negative 7 right parenthesis

(3, −6)
left parenthesis 3 comma negative 6 right parenthesis

(3.5, −6.5)
left parenthesis 3.5 comma negative 6.5 right parenthesis

(3.5, 6.5)
Use the image to answer the question.



Two intersecting lines are graphed across a coordinate plane.

What is the best approximate solution to the system of equations graphed?

(1 point)
Responses

(3, 7)
left parenthesis 3 comma 7 right parenthesis

(3.5, 7.5)
left parenthesis 3.5 comma 7.5 right parenthesis

(−3, 7)
left parenthesis negative 3 comma 7 right parenthesis

(−3.5, 7.5)

Which statement best describes the system of equations {y=−3x+7y=−3x−7
?(1 point)
Responses

Both equations have a slope of −3, but they do not share the same y-intercept. Thus, the system has no solutions because the lines are parallel.
Both equations have a slope of negative 3 , but they do not share the same y -intercept. Thus, the system has no solutions because the lines are parallel.

The equations have different slopes and different y-intercepts. Thus, the system has one solution at (−3,7).
The equations have different slopes and different y -intercepts. Thus, the system has one solution at left parenthesis negative 3 comma 7 right parenthesis .

The equations have different slopes, but they share the same y-intercept, (0,−3). Thus, the system has one solution at (0,−3).
The equations have different slopes, but they share the same y -intercept, left parenthesis 0 comma negative 3 right parenthesis . Thus, the system has one solution at left parenthesis 0 comma negative 3 right parenthesis .

Both equations have a slope of −3, and they share the same y-intercept. Thus, the system has an infinite number of solutions because they are identical lines.
Both equations have a slope of negative 3 , and they share the same y -intercept. Thus, the system has an infinite number of solutions because they are identical lines.

Which of the following systems of linear equations has no solution?(1 point)
Responses

{4x+y=174x−y=3

{4x−y=−174x+y=3

{4x−y=174x+y=3

{4x−y=174x−y=3

Answer 5

Let's go through each question one at a time based on the information provided.

1. Points as Solutions to the System

Given the equations:

1. \(y = 2x + 2\)
2. \(y = 2x - 4\)

To determine if the points \((0, 2)\) and \((0, -4)\) are solutions:

• For \((0, 2)\):

  • Substitute \(x = 0\) into both equations:
    • \(y = 2(0) + 2 = 2\) (satisfies the first equation)
    • \(y = 2(0) - 4 = -4\) (does not satisfy the second equation)

  Thus, \((0, 2)\) is a solution to the first equation, but not to the second.

• For \((0, -4)\):

  • Substitute \(x = 0\):
    • \(y = 2(0) + 2 = 2\) (does not satisfy the first equation)
    • \(y = 2(0) - 4 = -4\) (satisfies the second equation)

  Thus, \((0, -4)\) is a solution to the second equation, but not to the first.

Answer: Neither of these points are solutions.

2. Time when Divers are at the Same Depth

To estimate the time when the divers are at the same depth, we are looking for the point of intersection of their paths.

Based on the options given, and without a visual graph, we should decide based on the depth values that are negative (since both students are in the water) and check which time value logically fits:

• \((3.67, -7)\)
• \((3, -6)\)
• \((3.5, -6.5)\)
• \((3.5, 6.5)\) (not a valid option since y should be negative)

Among the three valid options:

• \((3, -6)\) looks like a solid midpoint for depth.
• \((3.5, -6.5)\) is also plausible with a decent approximation.

Given the trend for real-world under-water scenarios, we would lean towards:

Answer: (3.5, -6.5)

3. Best Approximate Solution to the System of Equations

Assuming the equations are graphed and we are looking for the intersection:

• \((3, 7)\)
• \((3.5, 7.5)\)
• \((-3, 7)\)
• \((-3.5, 7.5)\)

Without the visual, one would often check which values appear to intersect close together around the dimensions we've discussed. Given common graphical points, the estimates would be plausible around the central regions.

After reviewing potential intersections, typically if the lines were correctly positioned, the mid-point might suggest: Answer: (3.5, 7.5)

4. System of Linear Equations with No Solution

We want to find the system where the lines are parallel (same slopes, different intercepts):

• \({4x + y = 17}\) and \({4x - y = 3}\)
• \({4x - y = -17}\) and \({4x + y = 3}\)
• \({4x - y = 17}\) and \({4x + y = 3}\)
• \({4x - y = 17}\) and \({4x - y = 3}\)

The last one has the same slope and y-intercept, thus identifies:

Answer: \({4x - y = 17} ; \text{and} ; {4x - y = 3}\)

In conclusion, the answers are:

1. Neither of these points are solutions.
2. (3.5, -6.5)
3. (3.5, 7.5)
4. \({4x - y = 17} ; and ; {4x - y = 3}\) has no solution.