How to Solve a System of Equations

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Solving a system of simultaneous equations seems like a very daunting task at first. With more than one unknown quantity to find the value for, and apparently very little way of disentangling one variable from another, it can be a headache for people new to algebra. However, there are three different methods for finding the solution to the equation, with two depending more on algebra and being a bit more reliable, and the other turning the system into a series of lines on a graph.

Solving a System of Equations by Substitution

  1. Put One Variable in Terms of the Other

  2. Solve a system of simultaneous equations by substitution by first expressing one variable in terms of the other. Using these equations as an example:

    x y = 5

    3_x_ + 2_y_ = 5

    Re-arrange the simplest equation to work with and use this to insert into the second. In this case, adding y to both sides of the first equation gives:

    x = y + 5

  3. Substitute the New Expression Into the Other Equation

  4. Use the expression for x in the second equation to produce an equation with a single variable. In the example, this makes the second equation:

    3 × (y + 5) + 2_y_ = 5

    3_y_ + 15 + 2_y_ = 5

    Collect the like terms to get:

    5_y_ + 15 = 5

  5. Re-arrange and Solve for the First Variable

  6. Re-arrange and solve for y, starting by subtracting 15 from both sides:

    5_y_ = 5 – 15 = −10

    Dividing both sides by 5 gives:

    y = −10 ÷ 5 = −2

    So y = −2.

  7. Use Your Result to Find the Second Variable

  8. Insert this result into either equation to solve for the remaining variable. At the end of step 1, you found that:

    x = y + 5

    Use the value you found for y to get:

    x = −2 + 5 = 3

    So x = 3 and y = −2.

    Tips

    • Check Your Answers

      It’s good practice to always check that your answers make sense and work with the original equations. In this example, xy = 5, and the result gives 3 – (−2) = 5, or 3 + 2 = 5, which is correct. The second equation states: 3_x_ + 2_y_ = 5, and the result gives 3 × 3 + 2 × (−2) = 9 – 4 = 5, which is again correct. If something doesn’t match up at this stage, you have made a mistake in your algebra.

Solving a System of Equations by Elimination

  1. Choose a Variable to Eliminate and Adjust the Equations as Needed

  2. Look at your equations to find a variable to remove:

    x y = 5

    3_x_ + 2_y_ = 5

    In the example, you can see that one equation has -y and the other has +2_y_. If you add twice the first equation to the second one, the y terms would cancel out and y would be eliminated. In other cases (e.g., if you wanted to eliminate x), you can also subtract a multiple of one equation from the other.

    Multiply the first equation by two to prepare it for the elimination method:

    2 × (xy) = 2 × 5

    So

    2_x_ – 2_y_ = 10

  3. Eliminate One Variable and Solve for the Other

  4. Eliminate your chosen variable by adding or subtracting one equation from the other. In the example, add the new version of the first equation to the second equation to get:

    3_x_ + 2_y_ + (2_x_ – 2_y_) = 5 + 10

    3_x_ + 2_x_ + 2_y_ – 2_y_ = 15

    So this means:

    5_x_ = 15

    Solve for the remaining variable. In the example, divide both sides by 5 to get:

    x = 15 ÷ 5 = 3

    As before.

  5. Use Your Result to Find the Second Variable

  6. Like in the previous approach, when you have one variable, you can insert this into either expression and re-arrange to find the second. Using the second equation:

    3_x_ + 2_y_ = 5

    So, since x = 3:

    3 × 3 + 2_y_ = 5

    9 + 2_y_ = 5

    Subtract 9 from both sides to get:

    2_y_ = 5 – 9 = −4

    Finally, divide by two to get:

    y = −4 ÷ 2 = −2

Solving a System of Equations by Graphing

  1. Convert the Equations to Slope-Intercept Form

  2. Solve systems of equations with minimal algebra by graphing each equation and looking for the x and y value where the lines intersect. Convert each equation to slope-intercept form (y = mx + b) first.

    The first example equation is:

    x y = 5

    This can be converted easily. Add y to both sides and then subtract 5 from both sides to get:

    y = x – 5

    Which has a slope of m = 1 and a y-intercept of b = −5.

    The second equation is:

    3_x_ + 2_y_ = 5

    Subtract 3_x_ from both sides to get:

    2_y_ = −3_x_ + 5

    Then divide by 2 to get the slope-intercept form:

    y = −3_x_/2 + 5/2

    So this has a slope of m = -3/2 and a y-intercept of b = 5/2.

  3. Plot the Lines on a Graph

  4. Use the y intercept values and the slopes to plot both lines on a graph. The first equation crosses the y axis at y = −5, and the y value increases by 1 every time the x value increases by 1. This makes the line easy to draw.

    The second equation crosses the y axis at 5/2 = 2.5. It slopes downwards, and the y value decreases by 1.5 every time the x value increases by 1. You can calculate the y value for any point on the x axis using the equation if it’s easier.

  5. Find the Point of Intersection

  6. Locate the point where the lines intersect. This gives you both the x and y coordinates of the solution to the system of equations.

References

About the Author

Lee Johnson is a freelance writer and science enthusiast, with a passion for distilling complex concepts into simple, digestible language. He's written about science for several websites including eHow UK and WiseGeek, mainly covering physics and astronomy. He was also a science blogger for Elements Behavioral Health's blog network for five years. He studied physics at the Open University and graduated in 2018.

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