How to Find Linear Equations

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Linear equations form the basis of any Algebra I class, and students must understand them before they will be ready to move on to higher level algebra courses. Unfortunately, teachers and textbooks tend to break up the basics of linear equations into many fragmented ideas and skills that make the topic more confusing. If you can remember one basic formula called the "point-slope" formula, you will be able to tackle almost any question that asks you to solve a linear equation.

    Interpret the information given in the problem. This is the most difficult step. There are many different ways that the problem might give you the information (see tips below for examples), but it will give you either a slope and a coordinate point, or two coordinate points each for two points in a line.

    Calculate the slope (which is called "m") using your two points. The slope is the distance the line rises for every unit that it runs (or moves to the right). Subtract the y-coordinate (second number) of the second point from the y-coordinate of the first point. Divide this by the result of subtracting the x-coordinate (of the first point) of the second point from the x-coordinate of the second point. For example, if the coordinates of the first point are (2,2) (2 on each axis) and the coordinates of the second point are (3,4) (3 on the x-axis and 4 on the y-axis) then (4-2)/(3-2) = 2. For every space on your graph paper to the right, the line rises two spaces.

    Write down the slope and circle one of your points. It does not matter which one, but picking a point with a "0" or "1" in it will make your math work easier. From this step forward, you will no longer use the un-circled point.

    Use the slope and the point to fill in the point-slope formula which looks like this: y - y1 = m (x - x1).

    Look at the directions of the problem to see which form your linear equation should follow. If it asks for "point-slope" form, you are done. If it asks for "slope-intercept" formula, you will need to solve for "y" and simplify.

    Put the linear equation in slope-intercept formula y = mx + b (which is the form most useful for graphing), by solving for "y."


    • Some ways that a question might give you a slope/point or two points: 2 intercepts, a labeled graph picture showing two points or a point and a slope, information about parallel or perpendicular lines (which tells you about the slope), an intercept and the slope, 2 points, or statements that a line is horizontal or vertical.


    • Don't forget that subtracting a negative changes to addition. So if you have 3 - -4, you would end up with 7.

      Don't forget to distribute the negative sign when dealing with a negative slope.


About the Author

Kathryn White has over 11 years of experience tutoring a range of subjects at the kindergarten through college level. Her writing reflects her instructional ability as well as her belief in making all concepts understandable and approachable. White earned a Bachelor of Arts in history from Illinois Wesleyan University.

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