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.
slope: -3
y - intercept: 2

Explanation:

Part 1: Solve the system of equations

The system of equations is:
\[

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

\]

Step 1: Analyze the equations

Notice that both equations are identical. Let's rewrite the second equation: \( y = 2 - 3x \) is the same as \( y = -3x + 2 \) (by the commutative property of addition: \( a + b = b + a \), so \( 2 - 3x=-3x + 2 \)).

Step 2: Determine the solution

Since the two equations represent the same line, there are infinitely many solutions. All points \((x, y)\) that satisfy \( y=-3x + 2 \) are solutions. In set - builder notation, the solution is \(\{(x,y)\mid y=-3x + 2,x\in\mathbb{R}\}\)

Part 2: Identify \(m\) (slope) and \(b\) (y - intercept) for \(y = mx + b\)

For the equation \( y=-3x + 2 \), comparing with the slope - intercept form \( y=mx + b \):

• The slope \( m=-3 \) (the coefficient of \( x \)).
• The y - intercept \( b = 2 \) (the constant term).

Part 3: Graph the equation \(y=-3x + 2\)

1. Plot the y - intercept:

The y - intercept is \( b = 2 \), so we start by plotting the point \((0,2)\) on the coordinate plane (where \( x = 0 \) and \( y = 2 \)).

1. Use the slope to find another point:

The slope \( m=-3=\frac{-3}{1}=\frac{\text{rise}}{\text{run}}\). From the point \((0,2)\), we move down 3 units (because the rise is - 3) and then 1 unit to the right (because the run is 1). This gives us the point \((0 + 1,2-3)=(1,-1)\). We can also move up 3 units and left 1 unit from \((0,2)\) to get the point \((0 - 1,2 + 3)=(-1,5)\).

1. Draw the line:

Connect the points \((0,2)\), \((1,-1)\), \((-1,5)\) (and other points found using the slope) with a straight line. This line represents the equation \( y=-3x + 2 \).

Final Answers

• Solution of the system: Infinitely many solutions, \(\boldsymbol{\{(x,y)\mid y=-3x + 2,x\in\mathbb{R}\}}\)
• Slope (\(m\)): \(\boldsymbol{-3}\)
• y - intercept (\(b\)): \(\boldsymbol{2}\)
• Graph: A straight line passing through \((0,2)\), \((1,-1)\), \((-1,5)\) (and other points determined by the slope - intercept form) representing \(y=-3x + 2\)

Answer:

Part 1: Solve the system of equations

The system of equations is:
\[

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

\]

Step 1: Analyze the equations

Notice that both equations are identical. Let's rewrite the second equation: \( y = 2 - 3x \) is the same as \( y = -3x + 2 \) (by the commutative property of addition: \( a + b = b + a \), so \( 2 - 3x=-3x + 2 \)).

Step 2: Determine the solution

Since the two equations represent the same line, there are infinitely many solutions. All points \((x, y)\) that satisfy \( y=-3x + 2 \) are solutions. In set - builder notation, the solution is \(\{(x,y)\mid y=-3x + 2,x\in\mathbb{R}\}\)

Part 2: Identify \(m\) (slope) and \(b\) (y - intercept) for \(y = mx + b\)

For the equation \( y=-3x + 2 \), comparing with the slope - intercept form \( y=mx + b \):

• The slope \( m=-3 \) (the coefficient of \( x \)).
• The y - intercept \( b = 2 \) (the constant term).

Part 3: Graph the equation \(y=-3x + 2\)

1. Plot the y - intercept:

The y - intercept is \( b = 2 \), so we start by plotting the point \((0,2)\) on the coordinate plane (where \( x = 0 \) and \( y = 2 \)).

1. Use the slope to find another point:

The slope \( m=-3=\frac{-3}{1}=\frac{\text{rise}}{\text{run}}\). From the point \((0,2)\), we move down 3 units (because the rise is - 3) and then 1 unit to the right (because the run is 1). This gives us the point \((0 + 1,2-3)=(1,-1)\). We can also move up 3 units and left 1 unit from \((0,2)\) to get the point \((0 - 1,2 + 3)=(-1,5)\).

1. Draw the line:

Connect the points \((0,2)\), \((1,-1)\), \((-1,5)\) (and other points found using the slope) with a straight line. This line represents the equation \( y=-3x + 2 \).

Final Answers

• Solution of the system: Infinitely many solutions, \(\boldsymbol{\{(x,y)\mid y=-3x + 2,x\in\mathbb{R}\}}\)
• Slope (\(m\)): \(\boldsymbol{-3}\)
• y - intercept (\(b\)): \(\boldsymbol{2}\)
• Graph: A straight line passing through \((0,2)\), \((1,-1)\), \((-1,5)\) (and other points determined by the slope - intercept form) representing \(y=-3x + 2\)