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# How to find the equation for a line Finding the equation for a line is a common problem in geometry and trigonometry. There are two common situations where you are asked to find the equation for a line: either you get a point on the line and the line slope, or you get two points on the line. In either case, it is not difficult to find the equation for that line, provided you use the correct formula and work carefully.

## Step

### Calculate the equation with a point and the slope

1. Connect the slope for m in the formula yy 1
= m (xx 1 ). This is called the point-slope formula. The point slope formula uses the slope and coordinates of a point along the line to find the y-intersection. Use the slope instead of m in yy 1 = m (xx 1 ). • For example, if you know the slope of the line is 2, your formula will look like this: yy 1 = 2 (xx 1 ) .
2. Substitute x 1 and y 1 with the coordinates of the point. Use the coordinates you are given as (x 1 y 1 ). Put the numbers in the corresponding place on your formula before solving the equation. 3. Solve the formula for y to get the final slope intercept form. Follow the mathematical order of operations and the distributive property to remove the x term from parentheses. • In our example, you would first use the distributive property to get y-3 = 2x-8.
• Then add 3 to each page so that y is by itself.
• The final equation for a line in slope intercept form with a slope of 2 containing the point (4 3) is y = 2x-5 .

### Find the equation with two points

1. Find the slope with m = (y 2 -y 1 ) / (x 2 -x 1 ). The ordered pairs of the coordinates you have listed as (x, y). Use the first set of coordinates as (x 1 y 1 ) and the second set of (x 2 y 2 ). Connect the numbers to the formula m = (y 2 -y 1 ) / (x 2 -x 1 ) and solve for m . 2. Substitute m into the slope intercept with the slope you found. The slope intercept formula for a line is written as y = m x + b, where m is the slope and b is the y-intersection (the point on the y-axis where the line passes it). Connect the number you found for your slope instead of m . • In our example, the formula would read y = 1x + b or y = x + b when you change the slope value.
3. Substitute x and y for one of the points you know to solve for the y intercept. Choose one of the ordered pairs to put in the slope intercept form. Put the x value instead of x and y value instead of y. • In this example, if you chose (3, 8) as your coordinates, the formula would read 8 = 1 (3) + b .
4. Solve the equation for b. After connecting the x and y values ​​and the slope to the formula, find the value of b in the equation. Follow the order first before moving the rest of the numbers to the other side. Leave b on one side of the equation to solve it. • In our example, the formula currently reads 8 = 1 (3) + b. Multiply 1 and 3 together to get 8 = 3 + b. Since 3 is a positive number, subtract 3 from each side to isolate b. This leaves you with 5 = b, or b = 5.
5. Connect the slope and y- the intersection of the slope intercept form to end the equation. When you are done, connect the slope for m and the y intercept for b. After that, you have found the equation for the line. • For example, the equation for the line with points of (3, 8) and (7) 12) is y = 1x + 5 or simply y = x + 5 .

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