graph the following equations
equation 1: ( y = x + 2 )
equation 2: ( y = -2x - 4 )
equation 3: ( y = \frac{2}{3}x )
equation 4: ( y = -\frac{1}{4}x + 3 )

Question

Accepted Answer

# Explanation for Graphing Linear Equations (Using Slope - Intercept Form $y = mx + b$, where $m$ is slope, $b$ is y - intercept)

## Equation 1: $y=x + 2$
### Step 1: Identify slope ($m$) and y - intercept ($b$)
For $y=x + 2$, comparing with $y = mx + b$, we have $m = 1$ (slope) and $b=2$ (y - intercept).
### Step 2: Plot the y - intercept
The y - intercept is at the point $(0,2)$. So we mark the point $(0,2)$ on the coordinate plane.
### Step 3: Use the slope to find another point
The slope $m = 1=\frac{1}{1}$, which means for a change of $1$ unit in the $x$ - direction (run), we have a change of $1$ unit in the $y$ - direction (rise). From $(0,2)$, moving $1$ unit to the right (increase $x$ by $1$) and $1$ unit up (increase $y$ by $1$), we get the point $(1,3)$. We can also move $1$ unit to the left and $1$ unit down to get $(- 1,1)$.
### Step 4: Draw the line
Connect the points $(0,2)$, $(1,3)$ (or other points found) with a straight line.

## Equation 2: $y=-2x - 4$
### Step 1: Identify slope ($m$) and y - intercept ($b$)
For $y=-2x - 4$, comparing with $y = mx + b$, we have $m=-2$ (slope) and $b = - 4$ (y - intercept).
### Step 2: Plot the y - intercept
The y - intercept is at the point $(0,-4)$. Mark the point $(0,-4)$ on the coordinate plane.
### Step 3: Use the slope to find another point
The slope $m=-2=\frac{-2}{1}$, which means for a change of $1$ unit in the $x$ - direction (run), we have a change of $- 2$ units in the $y$ - direction (rise). From $(0,-4)$, moving $1$ unit to the right (increase $x$ by $1$) and $2$ units down (decrease $y$ by $2$), we get the point $(1,-6)$. We can also move $1$ unit to the left and $2$ units up to get $(-1,-2)$.
### Step 4: Draw the line
Connect the points $(0,-4)$, $(1,-6)$ (or other points found) with a straight line.

## Equation 3: $y=\frac{2}{3}x$
### Step 1: Identify slope ($m$) and y - intercept ($b$)
For $y=\frac{2}{3}x$, we can write it as $y=\frac{2}{3}x+0$. So $m = \frac{2}{3}$ (slope) and $b = 0$ (y - intercept).
### Step 2: Plot the y - intercept
The y - intercept is at the point $(0,0)$ (the origin). Mark the point $(0,0)$ on the coordinate plane.
### Step 3: Use the slope to find another point
The slope $m=\frac{2}{3}$, which means for a change of $3$ units in the $x$ - direction (run), we have a change of $2$ units in the $y$ - direction (rise). From $(0,0)$, moving $3$ units to the right (increase $x$ by $3$) and $2$ units up (increase $y$ by $2$), we get the point $(3,2)$. We can also move $3$ units to the left and $2$ units down to get $(-3,-2)$.
### Step 4: Draw the line
Connect the points $(0,0)$, $(3,2)$ (or other points found) with a straight line.

## Equation 4: $y=-\frac{1}{4}x + 3$
### Step 1: Identify slope ($m$) and y - intercept ($b$)
For $y=-\frac{1}{4}x + 3$, comparing with $y = mx + b$, we have $m=-\frac{1}{4}$ (slope) and $b = 3$ (y - intercept).
### Step 2: Plot the y - intercept
The y - intercept is at the point $(0,3)$. Mark the point $(0,3)$ on the coordinate plane.
### Step 3: Use the slope to find another point
The slope $m =-\frac{1}{4}=\frac{-1}{4}$, which means for a change of $4$ units in the $x$ - direction (run), we have a change of $- 1$ unit in the $y$ - direction (rise). From $(0,3)$, moving $4$ units to the right (increase $x$ by $4$) and $1$ unit down (decrease $y$ by $1$), we get the point $(4,2)$. We can also move $4$ units to the left and $1$ unit up to get $(-4,4)$.
### Step 4: Draw the line
Connect the points $(0,3)$, $(4,2)$ (or other points found) with a straight line.

# Answer:
To graph the equations:
- For $y = x+2$, plot $(0,2)$ and use slope $1$ to draw the line.
- For $y=-2x - 4$, plot $(0,-4)$ and use slope $-2$ to draw the line.
- For $y=\frac{2}{3}x$, plot $(0,0)$ and use slope $\frac{2}{3}$ to draw the line.
- For $y =-\frac{1}{4}x + 3$, plot $(0,3)$ and use slope $-\frac{1}{4}$ to draw the line.

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Question

Explanation:

Equation 1: \(y=x + 2\)

Step 1: Identify slope (\(m\)) and y - intercept (\(b\))

Step 2: Plot the y - intercept

Step 3: Use the slope to find another point

Step 4: Draw the line

Equation 2: \(y=-2x - 4\)

Step 1: Identify slope (\(m\)) and y - intercept (\(b\))

Step 2: Plot the y - intercept

Step 3: Use the slope to find another point

Step 4: Draw the line

Equation 3: \(y=\frac{2}{3}x\)

Step 1: Identify slope (\(m\)) and y - intercept (\(b\))

Step 2: Plot the y - intercept

Step 3: Use the slope to find another point

Step 4: Draw the line

Equation 4: \(y=-\frac{1}{4}x + 3\)

Step 1: Identify slope (\(m\)) and y - intercept (\(b\))

Step 2: Plot the y - intercept

Step 3: Use the slope to find another point

Answer: