Question

graph the line using the given points. then write an equation of the line.

1. (1,4) and (2,6)
2. (1,−2) and (2,−1)

write an equation of the line using the given two points.

1. goes through (−2,8) and (−1,5)

step 1: find slope step 2: substitute/solve for b step 3: write equation

1. goes through (−2,−6) and (−1,−2)

step 1: find slope step 2: substitute/solve for b step 3: write equation

1. goes through (3,−2) and (6,0)
2. goes through (16,5) and (32,9)

Explanation:

Problem 19: (1, 4) and (2, 6)

Step 1: Find Slope (\(m\))

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For points \((1,4)\) (\(x_1 = 1,y_1 = 4\)) and \((2,6)\) (\(x_2 = 2,y_2 = 6\)):
\(m=\frac{6 - 4}{2 - 1}=\frac{2}{1}=2\)

Step 2: Find \(b\) (y - intercept)

Use the slope - intercept form \(y=mx + b\). Substitute \(m = 2\), \(x = 1\), and \(y = 4\) into the equation:
\(4=2\times1 + b\)
\(4 = 2 + b\)
Subtract 2 from both sides: \(b=4 - 2=2\)

Step 3: Write Equation

Using \(y=mx + b\) with \(m = 2\) and \(b = 2\), the equation is \(y = 2x+2\)

Step 1: Find Slope (\(m\))

Using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\) with \((x_1,y_1)=(1,-2)\) and \((x_2,y_2)=(2,-1)\):
\(m=\frac{-1-(-2)}{2 - 1}=\frac{-1 + 2}{1}=1\)

Step 2: Find \(b\) (y - intercept)

Substitute \(m = 1\), \(x = 1\), and \(y=-2\) into \(y=mx + b\):
\(-2=1\times1 + b\)
\(-2=1 + b\)
Subtract 1 from both sides: \(b=-2 - 1=-3\)

Step 3: Write Equation

Using \(y=mx + b\) with \(m = 1\) and \(b=-3\), the equation is \(y=x - 3\)

Step 1: Find Slope

Using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\) with \((x_1,y_1)=(-2,8)\) and \((x_2,y_2)=(-1,5)\):
\(m=\frac{5 - 8}{-1-(-2)}=\frac{-3}{1}=-3\)

Step 2: Substitute/Solve for \(b\)

Substitute \(m=-3\), \(x=-2\), and \(y = 8\) into \(y=mx + b\):
\(8=-3\times(-2)+b\)
\(8 = 6 + b\)
Subtract 6 from both sides: \(b=8 - 6 = 2\)

Step 3: Write Equation

Using \(y=mx + b\) with \(m=-3\) and \(b = 2\), the equation is \(y=-3x + 2\)

Answer:

\(m = 2\), \(b = 2\), Equation: \(y = 2x + 2\)

Problem 20: (1, - 2) and (2, - 1)