what is the slope-intercept equation for (0,2) and (5,4)

Answer 1

The equation you are looking for is y = mx + b. You should memorize this equation as it is fundamental to math in any field, as it is the equation of a straight line. The variable y is what is on the y-axis (the vertical axis) and x is the value on the x-axis (the horizontal axis). These are your variables, and usually changing one will change the other. The slope, m, can be simplified to be the rise over run. In other words, slope is how far something moves on the y-axis divided by how far something moves on the x-axis. Finally, b is your y-intercept or where a line hits the y-axis.

First, you need to solve for m. As mentioned earlier, the slope (m) is rise over run, or y-axis distance divided by the x-axis distance. To determine the distance, you have to subtract the provided y and x variables. With the format of (x,y), we know that 0 and 5 are the x values and 2 and 4 are the y values. To determine slope, I will subtract the first set of numbers (0,2) from the second (5,4).

m = (y₂ - y₁) / (x₂ - x₁) = (4 - 2) / (5 - 0) = 2/5 = 0.4. This means our slope is 0.4. To see how we get this equation, see the bottom of this answer.

Now that we have slope, the intercept can be solved for. The easiest way to do this is to plug in and solve from y = mx + b. We can pick a y and x values, lets use (0,2), and solve.

2 = 0.4 * 0 + b which can be simplified to b = 2. So our y-intercept is 2. We can solve using the other x and y values to get the same answer and to check our work. If the answers do not match up, either our slope is wrong or we have a math error.

4 = 0.4*5 + b = 2 + b, subtracting 2 from each side, we end up with b = 2. Our answers agree, which helps confirm our slope and intercept we calculated.

Getting the slope equation

If we have the equation y₁ = mx₁ + b and y₂ = mx₂ + b, we have two unknowns (m and b), and two equations to solve. This means we can solve. If we rearrange the equations to be b = mx₁ - y₁ and b = mx₂ - y₂, we can set them equal to each other. The reason we can set them equal to each other is due to the y-intercept values (b) being the same number. This means we can get the equation...

mx₁ - y₁ = mx₂ - y₂

Now follow the steps below to see how we get the slope equation above.

y₂ + mx₁ - y₁ = mx₂

y₂ - y₁ = mx₂ - mx₁

y₂ - y₁ = m(x₂ - x₁)

(y₂ - y₁)/(x₂ - x₁) = m, which is our equation slope.