Question

solve the system of equations.
$y = -3x + 2$
$y = 2 - 3x$
$y = mx + b$
$m$: slope
$b$: $y$-intercept
graph the equation $y = -3x + 2$.

Explanation:

Step1: Identify the form of the equation

The equation \( y = -3x + 2 \) is in the slope - intercept form \( y=mx + b \), where \( m \) is the slope and \( b \) is the y - intercept.
For the equation \( y=-3x + 2 \), comparing with \( y = mx + b \), we have \( m=-3 \) (slope) and \( b = 2 \) (y - intercept).

Step2: Find the y - intercept point

The y - intercept is the point where the line crosses the y - axis. When \( x = 0 \), \( y=-3(0)+2=2 \). So the y - intercept point is \( (0,2) \).

Step3: Use the slope to find another point

The slope \( m=-3=\frac{\text{rise}}{\text{run}}=\frac{- 3}{1} \). Starting from the y - intercept point \( (0,2) \), to find the next point, we move down 3 units (because the rise is - 3) and 1 unit to the right (because the run is 1). So from \( (0,2) \), moving down 3 units gives \( y=2 - 3=-1 \) and moving 1 unit to the right gives \( x = 0+1 = 1 \). So the new point is \( (1,-1) \).

Step4: Graph the line

Plot the points \( (0,2) \) and \( (1,-1) \) (and we can find more points if needed, for example, using the slope again: from \( (1,-1) \), moving down 3 units (\( y=-1-3=-4 \)) and 1 unit to the right (\( x = 1 + 1=2 \)) gives \( (2,-4) \)) and then draw a straight line passing through these points.

To solve the system of equations \(

$$\begin{cases}y=-3x + 2\\y=2-3x\end{cases}$$

\):

Step1: Notice the equations

We can rewrite the second equation \( y = 2-3x \) as \( y=-3x + 2 \), which is the same as the first equation.

Step2: Determine the solution

Since the two equations represent the same line, the system of equations has infinitely many solutions. The solutions are all the points \( (x,y) \) that satisfy \( y=-3x + 2 \) (or \( y = 2-3x \)).

Answer:

• For the equation \( y=-3x + 2 \):
• Slope (\( m \)): \(-3\)
• y - intercept (\( b \)): \(2\)
• To graph: Plot \( (0,2) \), use slope \( \frac{-3}{1} \) to find other points (e.g., \( (1,-1) \), \( (2,-4) \)) and draw a line through them.
• For the system \(

$$\begin{cases}y=-3x + 2\\y=2-3x\end{cases}$$

\): Infinitely many solutions, all points on the line \( y=-3x + 2 \) (or \( y = 2-3x \)).