Transform the linear equation 4x−8y=24 to slope-intercept form.(1 point)

Question

Write y−2=3/2(x+2) in slope-intercept form.(1 point)

Graph the equation 9−x=3/2y using slope-intercept form.(1 point)

Answer 1

Let's address each part of your request step-by-step.

1. Transform the linear equation \(4x - 8y = 24\) to slope-intercept form.

The slope-intercept form of a linear equation is given by \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Starting with the equation: \[ 4x - 8y = 24 \]

First, isolate \(y\): \[ -8y = -4x + 24 \]

Now, divide every term by \(-8\): \[ y = \frac{1}{2}x - 3 \]

Thus, the slope-intercept form is: \[ y = \frac{1}{2}x - 3 \]

2. Write \(y - 2 = \frac{3}{2}(x + 2)\) in slope-intercept form.

Start with the equation: \[ y - 2 = \frac{3}{2}(x + 2) \]

Distribute \(\frac{3}{2}\) on the right side: \[ y - 2 = \frac{3}{2}x + 3 \]

Now, add \(2\) to both sides to solve for \(y\): \[ y = \frac{3}{2}x + 3 + 2 \] \[ y = \frac{3}{2}x + 5 \]

Thus, the slope-intercept form is: \[ y = \frac{3}{2}x + 5 \]

3. Graph the equation \(9 - x = \frac{3}{2}y\) using slope-intercept form.

Start with the equation: \[ 9 - x = \frac{3}{2}y \]

Rearranging to isolate \(y\): \[ \frac{3}{2}y = 9 - x \]

Multiply through by \(\frac{2}{3}\): \[ y = \frac{2}{3}(9 - x) \] \[ y = 6 - \frac{2}{3}x \]

Now, the slope-intercept form is: \[ y = -\frac{2}{3}x + 6 \]

Graphing the Equation

To graph \(y = -\frac{2}{3}x + 6\):

1. Identify the y-intercept: \(b = 6\). This is the point \((0, 6)\) on the y-axis.
2. Identify the slope: \(m = -\frac{2}{3}\). This means for every 3 units you move to the right (positive x-direction), you move down 2 units (negative y-direction).

Plotting the points:

• Start at the y-intercept \((0, 6)\).
• From \((0, 6)\), move 3 units to the right to \((3, 6)\) and then down 2 units to the point \((3, 4)\).
• Draw a straight line through these points, extending it in both directions.

Now you've transformed the equations and know how to graph the last one!

Answer 2

how did u get three when u distibuted the 2nd one

Answer 3

Let’s carefully go through the distribution step in the equation \(y - 2 = \frac{3}{2}(x + 2)\).

Starting Equation

\[ y - 2 = \frac{3}{2}(x + 2) \]

Step 1: Distributing \(\frac{3}{2}\)

When you distribute \(\frac{3}{2}\) across the quantity \((x + 2)\), you need to apply the multiplication separately to both \(x\) and \(2\):

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

Calculating each part:

1. \(\frac{3}{2} \cdot x = \frac{3}{2}x\)
2. \(\frac{3}{2} \cdot 2 = 3\) (since \(2 = \frac{2}{1}\) and \(\frac{3 \cdot 2}{2 \cdot 1} = \frac{6}{2} = 3\))

So the distribution outputs: \[ y - 2 = \frac{3}{2}x + 3 \]

Step 2: Solving for \(y\)

Now you add \(2\) to both sides to isolate \(y\):

\[ y = \frac{3}{2}x + 3 + 2 \]

Combine the constants: \[ y = \frac{3}{2}x + 5 \]

Final Result

The slope-intercept form is: \[ y = \frac{3}{2}x + 5 \]

Thank you for pointing out the need for clarity; I hope this explanation clarifies the distribution step!

Answer 4

graph 3x−y=10

Answer 5

To graph the equation \(3x - y = 10\), we'll first rewrite it in slope-intercept form, \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Step 1: Rewrite the equation

Starting with the original equation: \[ 3x - y = 10 \]

Isolate \(y\): \[ -y = -3x + 10 \] \[ y = 3x - 10 \]

Now we have it in slope-intercept form: \[ y = 3x - 10 \]

Step 2: Identify the slope and y-intercept

From the equation \(y = 3x - 10\):

• The slope \(m = 3\)
• The y-intercept \(b = -10\)

This means that the line crosses the y-axis at the point \((0, -10)\).

Step 3: Plot the y-intercept

1. Plot the point \((0, -10)\) on the Cartesian plane.

Step 4: Use the slope to find another point

Since the slope is \(3\) (or \(\frac{3}{1}\)), you can go up \(3\) units and then over \(1\) unit to the right:

From the point \((0, -10)\):

• Move up \(3\) to \(y = -7\), and move \(1\) unit right to \(x = 1\), giving you the point \((1, -7)\).

Step 5: Plot the second point

Plot the point \((1, -7)\).

Step 6: Draw the line

Now, draw a straight line through the points \((0, -10)\) and \((1, -7)\), extending the line in both directions. You can also find additional points if desired by using the slope again or using other values for \(x\).

Additional Points (Optional)

To find a few more points:

• For \(x = 2\): \[ y = 3(2) - 10 = 6 - 10 = -4 \quad \Rightarrow \quad (2, -4) \]
• For \(x = -2\): \[ y = 3(-2) - 10 = -6 - 10 = -16 \quad \Rightarrow \quad (-2, -16) \]

Final Result

The graph of the equation \(3x - y = 10\) is a straight line that crosses the y-axis at \((0, -10)\) and has a slope of \(3\), meaning it rises \(3\) units for every \(1\) unit it moves to the right.

If you were to visualize this on graph paper or using a graphing tool, you would see a line passing through those points.