Various ways to solve for a linear equation
Linear equations can be presented to you in various ways and by the same token you can solve a linear equation a number of ways. The first step in solving a linear equation is determining the slope.
Here is a list of ways to arrive at the slope.
1.) Given a graphed line, choose two points on the line. Count the rise on the y-axis over the run on the x-axis. Essentially, the slope, m, = rise/run.
2.) Starting with the coordinates of two points on line (x1, y1) and (x2, y2) use the formula: m = (y2 – y1) / (x2 – x1)
3.) Rewrite your equation in the form of y = mx + b. m, the coefficient of x, is your slope.
4.) A line with an unknown slope that is parallel to a line with slope m1, has the same slope as that line.
5.) A line with an unknown slope that is perpendicular to a line with slope m1 will have a slope equal to the opposite of the reciprocal of m1, that is -1/ m1.
Next, look at the data points that you have to work with. For example, if you are told that you have $150 to purchase bolts that cost $2 and washers that cost $3, then let x represent bolts and y represent washers. Therefore, your purchasing power is:
$150 = 2x + 3y. You can find two sets of coordinates by setting y = 0 and solving for x and x = 0 and solve for y, which will give you (75, 0) and (0, 50) respectively. Given these two points, you can solve for the slope using item number two, listed above.
m = (50 – 0) / (0 – 75) = -2/3
Once you calculate the slope, you will be ready to solve for your equation.
To do that, use the following method where cp = coordinate point and m=slope:
(y - cpy) / (x –cpx) = m
(y – 50) / (x -0) = -2/3
Steps to solve for the equation:
1.) (y – 50) / (x-0) = -2/3
2.) y – 50 = -2/3x
3.) y = -2/3x + 50
Voila, your equation!
Alternatively, and more even more quickly, you could have taken the original equation:
$150 = 2x + 3y
and taken the following steps to solve it:
1.) 3y = -2x + 150 (subtract 2x from both sides of the equation)
2.) y = -2/3x + 50 (divide out the three on both sides of the equation)
That will give you the slope intercept form y = -2/3x + 50
