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An ordered pair is always written (x,y)

.

Which ordered pair is a solution to the equation 5x+4y=6
?



(1 point)
Responses

(−2,4)
open paren negative 2 comma 4 close paren

(−2,1)
open paren negative 2 comma 1 close paren

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

(−1,2)

11 answers

To determine which ordered pair is a solution to the equation \( 5x + 4y = 6 \), we can substitute each pair into the equation and check if the left side equals the right side (6).

  1. For \((-2, 4)\): \[ 5(-2) + 4(4) = -10 + 16 = 6 \] This pair satisfies the equation.

  2. For \((-2, 1)\): \[ 5(-2) + 4(1) = -10 + 4 = -6 \] This does not satisfy the equation.

  3. For \((0, 2)\): \[ 5(0) + 4(2) = 0 + 8 = 8 \] This does not satisfy the equation.

  4. For \((-1, 2)\): \[ 5(-1) + 4(2) = -5 + 8 = 3 \] This does not satisfy the equation.

The only ordered pair that is a solution to the equation \( 5x + 4y = 6 \) is \((-2, 4)\).

Complete the ordered pair that is a solution for the equation x2−3y=1
(1 point)
(4,
)

To find the value of \( y \) in the ordered pair \( (4, y) \) that satisfies the equation \( x^2 - 3y = 1 \), we can substitute \( x = 4 \) into the equation:

\[ 4^2 - 3y = 1 \]

Calculating \( 4^2 \):

\[ 16 - 3y = 1 \]

Next, we can isolate \( y \) by solving the equation:

\[ 16 - 1 = 3y \] \[ 15 = 3y \] \[ y = \frac{15}{3} = 5 \]

So, the ordered pair that is a solution for the equation is \( (4, 5) \).

Iris wants to start saving money for new headphones. She starts with $44 already saved and decides to save an additional $15 per week. Write an equation that represents the total amount she has saved.

Use x
to represent the number of weeks and y
to represent the total amount she has saved.

(1 point)

To represent the total amount Iris has saved in terms of the number of weeks, we can write the equation as follows:

Let \( x \) represent the number of weeks, and \( y \) represent the total amount she has saved.

She starts with $44 and saves an additional $15 each week. The equation can be written as:

\[ y = 15x + 44 \]

This equation shows that the total amount saved \( y \) is equal to $15 times the number of weeks \( x \) plus her initial $44.

Select the inequality that is represented by the blue region of the graph.

(1 point)
Responses

y>3x−1
y is greater than 3 x minus 1

y<1−3x
y<1−3x

x>3y−1
x is greater than 3 y minus 1

y≤3x−1
y is less than or equal to 3 x minus 1
Question 2
Which of the following ordered pairs is a solution to the inequality?(1 point)
Responses

(0,−3)
open paren 0 comma negative 3 close paren

(3,1)
open paren 3 comma 1 close paren

(1,2)
open paren 1 comma 2 close paren

(0,0)

To determine which inequality corresponds to the blue region of the graph, we first need to understand the characteristics of the boundaries and the shading.

  1. Identifying the Boundary Line:

    • If the region is above the line represented by \( y = 3x - 1 \), then the inequality would be \( y > 3x - 1 \).
    • If the region is below the line, it would be \( y < 3x - 1 \).
    • If the region includes the line itself (solid line), it would involve \( \leq \) or \( \geq \).
    • If the line is dotted (not included), it would involve \( < \) or \( > \).

  2. Analysis of Given Inequalities:

    • \( y > 3x - 1 \)
    • \( y < 1 - 3x \)
    • \( x > 3y - 1 \)
    • \( y \leq 3x - 1 \)

Without the visual, I cannot definitively determine which inequality is correct. However, if the region is above the line with a solid boundary, the answer would be \( y \leq 3x - 1 \); if it is an open area above the line, then \( y > 3x - 1 \).

Question 2: Which of the following ordered pairs is a solution to the inequality?

To check which of the given ordered pairs satisfies the inequalities accurately, we'll need to substitute into the proposed inequalities. Let's check the four pairs against both \( y > 3x - 1 \) and \( y \leq 3x - 1 \) for example:

  1. For \( (0, -3) \): \[ -3 > 3(0) - 1 \quad \Rightarrow \quad -3 > -1 \quad \text{(False)} \] \[ -3 \leq 3(0) - 1 \quad \Rightarrow \quad -3 \leq -1 \quad \text{(False)} \]

  2. For \( (3, 1) \): \[ 1 > 3(3) - 1 \quad \Rightarrow \quad 1 > 8 \quad \text{(False)} \] \[ 1 \leq 3(3) - 1 \quad \Rightarrow \quad 1 \leq 8 \quad \text{(True)} \]

  3. For \( (1, 2) \): \[ 2 > 3(1) - 1 \quad \Rightarrow \quad 2 > 2 \quad \text{(False)} \] \[ 2 \leq 3(1) - 1 \quad \Rightarrow \quad 2 \leq 2 \quad \text{(True)} \]

  4. For \( (0, 0) \): \[ 0 > 3(0) - 1 \quad \Rightarrow \quad 0 > -1 \quad \text{(True)} \] \[ 0 \leq 3(0) - 1 \quad \Rightarrow \quad 0 \leq -1 \quad \text{(False)} \]

Conclusion for the Pairs:

  • \( (3, 1) \) and \( (1, 2) \) are solutions if the inequality was \( y \leq 3x - 1 \).
  • Depending on your graph (if it's above the line), only (0, 0) may be a solution if considering \( y > 3x - 1 \).

Choose based on the determined inequality from the graph and the points tested!

The graph shows the equation y=13x−2
in red and the equation y=−x+2
in blue.



(2 points)
What ordered pair is the solution of these two equations?

(
,
)

To find the ordered pair that is the solution of the equations \( y = 13x - 2 \) and \( y = -x + 2 \), we need to set the two equations equal to each other and solve for \( x \):

\[ 13x - 2 = -x + 2 \]

Now, let's rearrange this equation to isolate \( x \):

  1. Add \( x \) to both sides: \[ 13x + x - 2 = 2 \] \[ 14x - 2 = 2 \]

  2. Add 2 to both sides: \[ 14x = 4 \]

  3. Divide by 14: \[ x = \frac{4}{14} = \frac{2}{7} \]

Now that we have \( x \), we can substitute it back into either original equation to find \( y \). We'll use \( y = -x + 2 \):

\[ y = -\left(\frac{2}{7}\right) + 2 \] \[ y = -\frac{2}{7} + \frac{14}{7} = \frac{12}{7} \]

So, the ordered pair that is the solution of these two equations is:

\[ \left( \frac{2}{7}, \frac{12}{7} \right) \]

In decimal form, this pair would be approximately \( (0.29, 1.71) \).

The graph shows the equation y=x+2
in red and the equation y=x2
in blue.





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

(1 point)
Responses

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

(−1,1)
and (2,4)
open paren negative 1 comma 1 close paren and open paren 2 comma 4 close paren

(4,2)
open paren 4 comma 2 close paren

There are no solutions
There are no solutions

To find the solutions for the system of equations given by \( y = x + 2 \) (a linear equation) and \( y = x^2 \) (a quadratic equation), we need to set these equations equal to each other and solve for \( x \):

\[ x + 2 = x^2 \]

Rearranging the equation gives:

\[ x^2 - x - 2 = 0 \]

Now, we can factor this quadratic equation:

\[ (x - 2)(x + 1) = 0 \]

Setting each factor to zero gives us the solutions for \( x \):

  1. \( x - 2 = 0 \) → \( x = 2 \)
  2. \( x + 1 = 0 \) → \( x = -1 \)

Now we can find the corresponding \( y \) values for each \( x \):

  1. For \( x = 2 \): \[ y = 2 + 2 = 4 \quad \Rightarrow \quad (2, 4) \]

  2. For \( x = -1 \): \[ y = -1 + 2 = 1 \quad \Rightarrow \quad (-1, 1) \]

So the solutions to the system of equations are \( (-1, 1) \) and \( (2, 4) \).

Based on the provided options, the correct response is:

(−1,1) and (2,4).