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How Do You Put an Equation in Standard Form Into Slope-Intercept or Point-Slope Form?

Convert the following equation of a line in standard form to an equation in 1) slope-intercept form and 2) point-slope form. 2x+y=2, passing through the point (3,4)

Summary

  1. 2x-y=2 is in standard form
  2. (3,4) is a point on the line
  3. y=mx+b is slope-intercept form, where m is the slope and b is the y-intercept
  4. y=2x-2 is in slope-intercept form
  5. y-y1=m(x-x1) is point-slope form, where m is the slope and (x1,y1) is a point on the line
  6. y-4=2(x-3) is in point-slope form

Notes

    1. Right now our equation is in standard form
    2. Standard form is Ax+By=C, where A, B, and C are just numbers
    3. We want to put it in slope intercept form
    4. Slope-intercept form is y=mx+b, where 'm' is the slope and 'b' is the y-intercept
    1. Slope-intercept form is y=mx+b, where 'm' is the slope and 'b' is the y-intercept
    1. We want our equation to be in the form y=mx+b
    1. In slope-intercept form, y is by itself on one side of the equation
    1. Right now we have a 2x on the left hand side with our y
    2. Subtraction is the opposite of addition
    3. So we can move the 2x over to the other side by subtracting it from both sides
    1. This leaves us with just -y on the left hand side
    1. In slope-intercept form, we need a POSITIVE y by itself on one side
    1. Multiplying -y by -1 will flip the sign
    2. But remember, we need to do the same thing to both sides
    3. So we'll also multiply the right hand side by -1
    1. -y•-1=y
    1. Multiply -1 by each term in the parentheses on the right hand side
    2. 2•-1=-2
    3. -2x•-1=2x
    1. Remember, slope-intercept form is of the form y=mx+b
    1. The commutative property of addition says that it doesn't matter what order we add in
    2. We can change the order of our terms and get the same answer
    1. Our equation is now y=2x-2
    2. So m=2 and b=-2
    3. This means the slope is 2 and the y-intercept is -2
    1. Point-slope form is y-y1=m(x-x1)
    2. m is the slope
    3. (x1, y1) is a point on the line
    1. (x1, y1) is a point on the line
    1. We were given the point (3,4)
    2. So x1 is 3 and y1 is 4
    1. x1 is 3 and y1 is 4
    1. We were given the point (3,4)
    1. y1 is 4
    1. m is the slope, which we haven't figured out yet
    2. But we know that x1 is 3
    1. When we put our equation into slope-intercept form, we put it into a form where it's easy to pick out the slope
    2. Slope-intercept form is y=mx+b, where m is the slope and b is the y-intercept
    3. So the slope is just the number in front of the x, which is 2
    1. Slope-intercept form is y=mx+b, where m is the slope and b is the y-intercept
    2. So the slope is just the number in front of the x, which is 2
    1. Since we're working with the same equation, we can use the same slope
    1. You can plug any point in for (x1,y1), as long as you know it's on the line
    1. You can plug any point in for (x1,y1), as long as you know it's on the line
    2. If we then converted back into slope-intercept form, we'd get the same equation
    1. You can plug any point in for (x1,y1), as long as you know it's on the line
    1. The numbers in slope-intercept form represent the slope and the y-intercept
    2. The slope is always the same, and there is only 1 y-intercept
    3. So there is only one form of the equation in slope-intercept form