Question

y=(1/2x + 2)
slope__ y intercept__
y= 2x - 1
x | y
-2 | -5
-1 | -3
0 | -1
1 | 1
2 | 3
slope 2 y intercept -1
y=
x | y
-5 | 10
-4 | 8
-3 | 6
-2 | 4
-1 | 2
slope__ y intercept__

Explanation:

Step1: Find the slope

The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Let's take two points from the table, say \((-5, 10)\) and \((-4, 8)\). Then \( m=\frac{8 - 10}{-4 - (-5)}=\frac{-2}{1}=- 2 \). We can check with another pair, like \((-4, 8)\) and \((-3, 6)\): \( m=\frac{6 - 8}{-3 - (-4)}=\frac{-2}{1}=-2 \). So the slope is \(-2\).

Step2: Find the equation of the line

Using the slope - intercept form \( y=mx + b \), where \( m=-2 \). We can use a point, say \((-1, 2)\) (from the table: when \( x = - 1\), \( y = 2\)). Substitute into the equation: \( 2=-2\times(-1)+b \), which simplifies to \( 2 = 2 + b \). Subtract 2 from both sides: \( b=0 \). Wait, let's check with \( x = 0 \). Wait, in the table, when \( x=-5,y = 10\), \( x=-4,y = 8\), \( x=-3,y = 6\), \( x=-2,y = 4\), \( x=-1,y = 2\). If we extend the table to \( x = 0 \), when \( x = 0 \), using the slope \( m=-2 \), from \( x=-1,y = 2 \), when \( x\) increases by 1 (to \( x = 0\)), \( y\) decreases by 2, so \( y=2-2 = 0\). Wait, but let's re - calculate the \( y\) - intercept. The slope - intercept form is \( y=mx + b \). We know \( m=-2 \), let's use the point \((-1,2)\): \( 2=-2\times(-1)+b\Rightarrow2 = 2 + b\Rightarrow b = 0\). Wait, but when \( x = 0 \), \( y = 0\)? Wait, let's check the pattern. The \( y\) - values are decreasing by 2 as \( x\) increases by 1. When \( x=-1,y = 2\); \( x = 0,y=0\); \( x = 1,y=-2\) etc. Wait, but let's use the slope formula for the \( y\) - intercept. The \( y\) - intercept is the value of \( y\) when \( x = 0 \). From the table, we can see the pattern: \( y\) is related to \( x\) by \( y=-2x\) (since when \( x=-5,y = 10=-2\times(-5)\); \( x=-4,y = 8=-2\times(-4)\); \( x=-1,y = 2=-2\times(-1)\)). So when \( x = 0 \), \( y=-2\times0 = 0\). So the \( y\) - intercept \( b = 0\).

Step3: Confirm the slope and \( y\) - intercept

We found that the slope \( m=-2 \) and the \( y\) - intercept \( b = 0\). Let's verify with the equation \( y=-2x\). For \( x=-5\), \( y=-2\times(-5)=10\) (matches the table). For \( x=-1\), \( y=-2\times(-1)=2\) (matches the table). So the slope is \(-2\) and the \( y\) - intercept is \( 0\).

Answer:

Slope: \(-2\), \( y\) - intercept: \( 0\)