Question

practice problems

1. what is an equation for the line that passes through the coordinates (2,7) and (0, 1)?

1. what is an equation for the line that passes through the coordinates (2,0) and (0,3)?

1. what is an equation for the line that passes through the coordinates (-1,2) and (7,6)?

1. find the equation of the line that passes through the points (1,1) and (3,5)?

1. find the equation of the line that passes through the points (1,3) and (2,4)?

Explanation:

Problem 1

Step1: Find the slope ($m$)

The formula for slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For points \((2,7)\) and \((0,1)\), \(x_1 = 2,y_1 = 7,x_2=0,y_2 = 1\). So \(m=\frac{1 - 7}{0 - 2}=\frac{-6}{-2}=3\).

Step2: Use point - slope form \(y - y_1=m(x - x_1)\), we can also use the y - intercept form \(y=mx + b\). We know that when \(x = 0\) (from the point \((0,1)\)), \(y = 1\), so the y - intercept \(b = 1\).

The equation of the line is \(y=3x + 1\).

Step1: Calculate the slope ($m$)

Using the formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\) for points \((2,0)\) and \((0,3)\), where \(x_1 = 2,y_1 = 0,x_2 = 0,y_2=3\). Then \(m=\frac{3 - 0}{0 - 2}=\frac{3}{-2}=-\frac{3}{2}\).

Step2: Find the y - intercept ($b$)

We know that when \(x = 0\) (from the point \((0,3)\)), \(y = 3\), so \(b = 3\).
The equation of the line is \(y=-\frac{3}{2}x + 3\).

Step1: Determine the slope ($m$)

For points \((-1,2)\) and \((7,6)\), using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\), where \(x_1=-1,y_1 = 2,x_2 = 7,y_2=6\). So \(m=\frac{6 - 2}{7-(-1)}=\frac{4}{8}=\frac{1}{2}\).

Step2: Use point - slope form. Let's use the point \((-1,2)\). The point - slope form is \(y - y_1=m(x - x_1)\). Substituting \(m=\frac{1}{2},x_1=-1,y_1 = 2\), we get \(y - 2=\frac{1}{2}(x + 1)\).

Simplify it: \(y-2=\frac{1}{2}x+\frac{1}{2}\), then \(y=\frac{1}{2}x+\frac{1}{2}+2=\frac{1}{2}x+\frac{5}{2}\).

Answer:

\(y = 3x+1\)

Problem 2