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

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

Summary

  1. y=2x-2 is in slope-intercept form
  2. (3,4) is a point on the line y=2x-2
  3. Ax+By=C is standard form, where A, B, and C are numbers and A is positive
  4. 2x-y=2 is in standard 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. Standard form of a linear equation is Ax+By=C, where A, B and C are numbers and A is positive
    1. Standard form of a linear equation is Ax+By=C, where A, B and C are numbers and A is positive
    1. Standard form of a linear equation is Ax+By=C, where A, B and C are numbers and A is positive
    1. Right now our equation is in slope-intercept form
    2. We want it to be in standard form
    1. Standard form of a linear equation is Ax+By=C
    1. Right now our equation is in slope-intercept form
    2. The general form of slope-intercept form is y=mx+b, where m is the slope and b is the y-intercept
    1. A, the number in front of the x in the standard form equation, has to be positive
    2. So since 2, the number we have in front of the x, is already positive, we shouldn't try to do anything to it
    1. Remember, we want to get 'x' and 'y' on one side and a number on the other side
    2. Since we're not moving 'x', we need to get the 2 to the other side
    1. Addition is the opposite of subtraction
    2. So we can add 2 to both sides to undo the subtraction
    1. We can add 2 to both sides to undo the subtraction
    1. We can add 2 to both sides to undo the subtraction
    2. The 2's will cancel on the right, leaving us with 2x by itself
    1. Now we have our x term by itself on one side
    2. But we need our x and y terms on the SAME side
    1. Remember, in standard form, x and y are on the same side of the equation
    1. We can get x and y on the same side by subtracting y from both sides
    2. Subtraction is the opposite of addition
    1. We can subtract y from both sides to undo the addition
    2. The y's will cancel on the left, leaving us with just 2
    3. Now we have x and y together on one side!
    1. We have x and y together on one side and a number on the other
    2. The equation is almost in standard form!
    1. 2=2x+y is the same as 2x+y=2
    2. So we can flip our equation around so it looks like the general form for standard form
    1. Now our equation is in standard form!
    2. Remember, standard form is Ax+By=C
    3. So here A=2, B=-1, and C=2
    1. Remember, standard form is Ax+By=C
    1. Point-slope form of a linear equation is y-y1=m(x-x1), where m is the slope and (x1,y1) is a point on the line
    1. Point-slope form of a linear equation is y-y1=m(x-x1), where m is the slope and (x1,y1) is a point on the line
    1. Point-slope form of a linear equation is y-y1=m(x-x1), where m is the slope and (x1,y1) is a point on the line
    1. If we have the slope and a point, all we have to do is plug them into the formula to get our equation
    2. m is the slope
    3. (x1,y1) is any point on the line
    1. We already know that our line goes through the point (3,4)
    1. We can plug (3,4) in for (x1,y1) in the formula for point-slope form
    2. So we will plug in 3 for x1 and 4 for y1
    1. All you need is a point that you know is on the line
    2. So if you pick a value for x and plug it into the equation of the line you have, you can find the y value that goes with that x
    3. Then you can use that point to plug in for (x1,y1) in point-slope form
    1. 4 is the y-coordinate of the point we were given, so we plug it in for y1
    1. 3 is the x-coordinate of the point we were given, so we plug it in for x1
    2. m is the slope, but we don't know that yet
    1. Our last missing piece is m, the slope
    1. Slope-intercept has the form y=mx+b, where m is the slope and b is the y-intercept
    2. Our original equation was y=2x-2
    3. So m, the slope, is 2
    4. And b, the y-intercept, is -2
    1. Our original equation was y=2x-2
    2. So m, the slope, is 2
    3. And b, the y-intercept, is -2
    1. We know that m is the slope, which is 2
    2. Since we are working with the same line, we will have the same slope no matter what form the equation is in
    1. We can use this value for the slope in our point-slope equation
    1. Remember, our slope is 2
    2. So we can just plug 2 in for m into our point-slope equation
    1. Plug 2 in for m
    2. This gives us our equation in point-slope form!
    1. Our equation in point-slope form is y-4=2(x-3)
    1. Point-slope form is pretty easy if you know the slope and a point!