Question

slope

1. $y = -x + 2$
2. $y = \frac{4}{3}x - 4$
3. $y = 4$

Explanation:

To determine the slope of each line, we use the slope - intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept. We can also calculate the slope using the formula \(m=\frac{\text{rise}}{\text{run}}=\frac{y_2 - y_1}{x_2 - x_1}\) for two points \((x_1,y_1)\) and \((x_2,y_2)\) on the line.

Problem 2: \(y=-x + 2\)

Step 1: Identify the slope from the slope - intercept form

The equation of the line is in the form \(y=mx + b\), where \(m\) is the slope and \(b\) is the y - intercept. For the equation \(y=-x + 2\), we can rewrite \(-x\) as \(- 1x\). So, by comparing with \(y = mx + b\), we have \(m=-1\).
We can also verify this using two points on the line. Let's take two points: when \(x = 0\), \(y=2\) (the y - intercept \((0,2)\)), and when \(x = 2\), \(y=-2 + 2=0\) (the point \((2,0)\)).
Using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\), with \((x_1,y_1)=(0,2)\) and \((x_2,y_2)=(2,0)\), we get \(m=\frac{0 - 2}{2-0}=\frac{-2}{2}=-1\).

Problem 4: \(y=\frac{4}{3}x-4\)

Step 1: Identify the slope from the slope - intercept form

The equation \(y = \frac{4}{3}x-4\) is in the form \(y=mx + b\). By comparing, we can see that \(m = \frac{4}{3}\).
We can verify this with two points. When \(x = 0\), \(y=-4\) (the point \((0, - 4)\)), and when \(x = 3\), \(y=\frac{4}{3}\times3-4=4 - 4 = 0\) (the point \((3,0)\)).
Using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\), with \((x_1,y_1)=(0,-4)\) and \((x_2,y_2)=(3,0)\), we have \(m=\frac{0-(-4)}{3 - 0}=\frac{4}{3}\).

Problem 6: \(y = 4\)

Step 1: Analyze the line

The equation \(y = 4\) represents a horizontal line. For a horizontal line, the \(y\) - value is constant for all \(x\) - values. Let's take two points on the line, say \((0,4)\) and \((2,4)\).
Using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\), with \((x_1,y_1)=(0,4)\) and \((x_2,y_2)=(2,4)\), we get \(m=\frac{4 - 4}{2-0}=\frac{0}{2}=0\).
We can also note that in the slope - intercept form \(y=mx + b\), \(y = 0x+4\), so the slope \(m = 0\).

Answer:

s:

• For \(y=-x + 2\), the slope is \(\boldsymbol{-1}\).
• For \(y=\frac{4}{3}x-4\), the slope is \(\boldsymbol{\frac{4}{3}}\).
• For \(y = 4\), the slope is \(\boldsymbol{0}\).