Today's soundtrack is Years & Years: Communion.
This evening, I'll be focusing on three of the linear equations: Slope-Intercept form, Slope-Point form, and General form. This post will begin by defining the forms; by the end, we'll learn how to convert between each of the forms.
Forms and Examples
The first form, Slope-Intercept form, is shown by the equation y = mx + b. We use the Slope-Intercept form if we know the slope and the y-intercept (citation). We can use this form to describe any linear function (Pearson, Foundations of Math and Pre-Calculus 10). Y refers to the Y-axis; M refers to the slope; X refers to the X-axis, and B refers to the Y-intercept.
We would use this form to calculate the cost of an appliance servicing. If we know that the call-out cost is $50, and the technician charges $100 per hour, we could plug those numbers into "y = mx + b" say that the equation is Y = 100/1(x) + 50. Using that equation, we can determine the cost (Y) of a servicing that takes three hours (x): Y = 100/1(3) +50; therefore, the cost of a three-hour service is $350.
The second form, Slope-Point form, is shown by the equation y - y1 = m (x - x1). We use this form if we know the slope and any point of the line (citation). To fill in the equation, we just plug in the coordinates of the point that we know into the y1 and x1 part of the equation, and input the slope where "m" is. For example, if we know that a line has the slope (3/2) and goes through the point (4, -7), we would say y - -7 = 3/2 (x - 4). Since we are subtracting a negative on the Y axis, we switch the symbol and instead say y + 7 = 3/2 (x - 4), the final form of our equation.
We can also use this form to determine both the x and y intercepts by solving for 0: in that same equation, y + 7 = 3/2 (x - 4), if we wanted to find the x-intercept, we would just say that y = 0 (because the x-intercept will be on the y axis at 0), so we would solve for x in the equation 0 + 7 = 3/2 (x - 4). Our first step would be to use distributive properties to get rid of the bracket, multiplying 3/2 by both x and -4, giving us 7 = 3/2x - 12/2. Then we flip the sides so that we can isolate for x: 3/2x - 12/2 = 7. Next let's start moving the equation around: 3/2x - 12/2 = 14/2+12/2; 3/2x = 26/2; divide both sides of the equation to isolate x: x = (26/2)/(3/2); we then flip 3/2 to 2/3 and multiply that by 26/2 to divide 26/2 by 3/2, giving us 26/3, an improper fraction that breaks down to a mixed fraction of 8 2/3; therefore, our x-intercept is 8.67. If we wanted to find the y-intercept, we would follow the same steps, solving of course for y + 7 = 3/2 (0 - 4).
The third form, called General form, is shown by the equation Ax + By + C = 0. In this form, A represents a whole number; B and C are integers. The only benefit of using the General form is that it can express vertical and vertical lines well (citation).
Converting Between Forms (with examples)
Slope intercept to slope point (y = mx + b) to [y-y1=m(x-x1)]
Start with the slope intercept equation y = -1/2x + 2
We know that "+2" is the y-int; therefore, our "x" is 0; we just plug those numbers into the slope point equation: y - 2 = -1/2 (x - 0) is our final form!
Slope intercept to general equation (y = mx + b) to (Ax + By + C = 0)
Start with the slope intercept equation y = -1/2x + 2
First, we want to get our variables onto the same side of the equation, so we say y + 1/2x = -1/2x + 1/2x + 2; therefore 1/2x + y = 2
Next, we want to get the constant onto the other side: 1/2x + y - 2 = 2 - 2; 1/2x + y -2 = 0 - our final form!
Slope point to slope intercept [y-y1=m(x-x1)] to (y = mx + b)
Start with the slope point equation y - 4 = 3 (x + 1)
First, we use distributive properties to remove the brackets, so we multiply 3 by both x and 1: y - 4 = 3x + 3
We need to isolate y, so we say y - 4 + 4 = 3x + 3 + 4; this gives us y = 3x +7, our final form!
Slope point to general equation [y-y1=m(x-x1)] (to (Ax + By + C = 0)
Start with the slope point equation y - 4 = 3 (x + 1)
First, we use distributive properties to remove the brackets, so we multiply 3 by both x and 1: y - 4 = 3x + 3
Next, we want to get everything to the left side of the equation: y - 4 - 3 = 3x + 3 - 3; y - 1 = 3x; -3x + y - 7 = 0. We are nearly there! We never want to start our equation with a negative, so divide everything by -1, giving us 3x - y + 7, which is our final form!
General equation to slope intercept (Ax + By + C = 0) to (y = mx + b)
Start with the general form equation x - 3y + 9 = 0
First, we want to isolate y, so we say x - 3y + 3y + 9 = 0 + 3y, which gives us x + 9 = 3y
Next, we flip the equation so that it's in the right order: 3y = x + 9
Now we divide everything by the coefficient of y, because the slope intercept form boils down to y equaling 0: 3y/3 = x/3 + 9/3, which gives us y = 1/3x + 3, which means we've reached our final form!
General equation to slope point (Ax + By + C = 0) to [y-y1=m(x-x1)]
Start with the general form equation x - 3y + 9 = 0
First, we want to isolate y, so we say x - 3y + 3y + 9 = 0 + 3y, which gives us x + 9 = 3y
Next, divide everything by the coefficient of y: x/3 + 9/3 = y, then simplify: 1/3x + 3 = y.
Flip the equation around; it becomes y = 1/3x + 3
Use 0 to represent x and solve for y: y = 3
Input the data into the slope point form equation: y - 3 = 1/3 (x - 0)
And there you have it! Hope this helped you as much as it helped me.
