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Home / 1.1.3 Quiz Finding Equations of Lines as Solutions

1.1.3 Quiz Finding Equations of Lines as Solutions

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Finding the equation for a line is a common problem in geometry and trigonometry. There are two common situations where you are asked to find the equation for a line: either you'll be provided with one point on the line and the slope of the line, or you'll be provided two points on the line. In either case, finding the equation for that line isn't difficult, provided you use the correct formula and work carefully.

  1. 1

    Plug the slope in for m in the formula y-y1 = m(x-x1). This is known as the point-slope formula.[1] The point-slope formula uses the slope and the coordinates of a point along the line to find the y-intercept. Use the slope in place of m in y-y1 = m(x-x1).[2]

    • For example, if you know the slope of the line is 2, then your formula will look like this: y-y1 = 2(x-x1).

  2. 2

    Replace x1 and y1 with the coordinates of the point. Use the coordinates you're given as (x1, y1). Put the numbers in the corresponding spot on your formula before you start solving the equation.[3]

    For example, if you know the coordinate is (4, 3), your formula will read: y-3 = 2(x-4).

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  3. 3

    Solve the formula for y to get the final slope-intercept formula. Follow the mathematical order of operations and the distributive property to remove the x-term from parenthesis.[4]

    • In our example, first you'd use the distributive property to get y-3=2x-8.
    • Then, add 3 to each side so y is by itself.
    • The final equation for a line in slope-intercept form with a slope of 2 that contains the point (4, 3) is y = 2x-5.

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  1. 1

    Find the slope using m = (y2-y1)/(x2-x1). The ordered pairs of the coordinates you have are listed as (x, y). Use the first set of coordinates as (x1, y1) and the second set as (x2, y2). Plug the numbers into the formula m = (y2-y1)/(x2-x1) and solve for m.[5]

    For example, if your coordinates are (3, 8) and (7, 12), the formula would read: m = (12-8)/(7-3) = 4/4 = 1. In this case, your slope, or m, equals 1.

  2. 2

    Replace the m in the slope-intercept formula with the slope you found. The slope-intercept formula of a line is written as y = mx+b, where m is the slope and b is the y-intercept (the point on the y-axis where the line crosses it). Plug the number you found for your slope in place of m.[6]

    • In our example, the formula would read y = 1x+b or y = x+b when you replace the slope value.

  3. 3

    Substitute x and y for one of the points you know to solve for the y-intercept. Pick one of the ordered pairs to put into the slope-intercept formula. Put the x-value in place of x and the y-value in place of y.[7]

    • In this example, if you chose (3, 8) as your coordinates, then the formula would read 8 = 1(3)+b.

  4. 4

    Solve the equation for b. Once you plug the x- and y-values as well as your slope into the formula, find the value of b in the equation. Follow the order of operations first before moving the rest of the numbers to the other side. Leave b on one side of the equation to solve it.[8]

    • In our example, the formula currently reads 8 = 1(3)+b. Multiply 1 and 3 together to get 8 = 3+b. Since 3 is a positive number, subtract 3 from each side to isolate b. This leaves you with 5 = b, or b = 5.

  5. 5

    Plug in the slope and y-intercept into the slope-intercept formula to finish the equation. Once you're finished, plug in the slope for m and the y-intercept for b. After that, you've found the equation for the line.

    • For example, the equation for the line with points on (3, 8) and (7, 12) is y = 1x+5 or simply y = x+5.

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Add New Question

  • Question

    How do I find an intercept of a line?

    Donagan

    Using the equation, set x equal to zero and solve for y to find the y-intercept, or set y equal to zero and solve for x to find the x-intercept.

  • Question

    What should I do with a negative y?

    Community Answer

    You multiply the whole equation by -1 to remove the negative sign. For example, if the question is: -y=-5x+1, you would then change the question to: -y*(-1) = (-5x+1)*(-1), and then get: y=5x-1.

  • Question

    What about given a slope and an intercept in y?

    Donagan

    Express the equation in standard form, y = mx + b, where m is the slope, and b is the y-intercept.

  • Question

    The teacher wants me to write the equation of a line. If the gradient is 3 and the y intercept is -5, is the equation y = 3x + -5?

    Donagan

    Yes, y = 3x - 5.

  • Question

    What if my parallel line is the y-axis?

    Donagan

    That means the line you're looking for would be defined as x = b: the line is vertical, m is "undefined" (infinity), and b is the x-intercept.

  • Question

    How do I find whether a point (coordinate) falls on a line?

    Donagan

    Take the x- and y-coordinates of the point, and insert them into the equation of the line. If the equation holds true with those x and y values inserted, the point is on the line.

  • Question

    How do I find a parallel line?

    Donagan

    Assuming a given straight line in the form of y = mx + b, any parallel line would have the same form with the same "m" but a different "b."

  • Question

    How would I solve a system with only one given point?

    Donagan

    A "system" consists of at least two different equations (and thus at least two lines on a graph). As the above article explains, two pieces of information are required in order to define a line: a point and a slope. To "solve a system with only one given point," the point would have to be the intersection of the lines, and you would then need each line's slope.

  • Question

    What about one point and no slope?

    Donagan

    As the above article explains, two pieces of information are required in order to define a straight line: the line's slope and a point on the line. If you don't know the slope, you could have an infinite number of lines passing through a given point.

  • Question

    How to find the equation of a line when points are given?

    Donagan

    Use Method 2 above.

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Article SummaryX

To find the equation of a line using 2 points, start by finding the slope of the line by plugging the 2 sets of coordinates into the formula for slope. Then, plug the slope into the slope-intercept formula, or y = mx + b, where "m" is the slope and "x" and "y" are one set of coordinates on the line. Next, solve the formula to find the value of "b," which is the y-intercept. Finally, plug the slope and y-intercept into the slope-intercept formula to finish the equation of the line. To learn how to find the equation of a line using 1 point, scroll down!

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1.1.3 Quiz Finding Equations of Lines as Solutions

Source: https://www.wikihow.com/Find-the-Equation-of-a-Line