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For “system of equations” you will have to solve two or more equations at the same time. When these equations have two different variables, such as x and y or a and b, you may not know how to solve them at first. Luckily, all you need is basic algebra skills to solve a system of equations (sometimes a little knowledge of fractions is required). If you are in the habit of learning visually, or if your teacher requires it, you should learn how to graph equations. Graphing can be a useful way to “see what’s going on” or to check results, but this is slower than other solutions and doesn’t work for every system of equations.
Steps
Use the alternative method
- This method often requires the use of fractions in the solution. You can use the elimination method below if you don’t like using fractions.
- 4x = 8 – 2y
- (4x)/4 = (8/4) – (2y/4)
- x = 2 – y
- You know that x = 2 – ½y .
- The second equation that hasn’t been transformed is 5x + 3y = 9 .
- In the second equation, replace x with “2 – ½y”: 5(2 – ½y) + 3y = 9 .
- 5(2 – y) + 3y = 9
- 10 – (5/2)y + 3y = 9
- 10 – (5/2)y + (6/2)y = 9 (If you don’t understand this step, learn how to add fractions. This is common knowledge in the substitution method, but not always. .)
- 10 + y = 9
- y = -1
- y = -2
- You know that y = -2
- One of the two original equations is 4x + 2y = 8 . (You can use either equation at this step.)
- Substitute -2 for y: 4x + 2(-2) = 8 .
- 4x – 4 = 8
- 4x = 12
- x = 3
- If you have an equation that contains no variables and is incorrect (eg 3 = 5), then the problem has no solution . (If you graph the two equations, you’ll see that they’re parallel and never intersect.)
- If you have an equation that contains no variables but holds true (for example, 3 = 3), then the problem has infinitely many solutions . These two equations are exactly the same. (If you graph the two equations, you’ll see that they are a single line.)
Using the suppression method
- You have a system of equations 3x – y = 3 and -x + 2y = 4 .
- Let’s transform the first equation so that the y variable cancels out. (You can choose the variable x and still end up with the same answer.)
- The -y variable in the first equation needs to be suppressed with + 2y in the second equation. This can happen if you multiply -y by 2.
- Multiply both sides of the first equation by 2: 2(3x – y)=2(3) , so 6x – 2y = 6 . Now – 2y will cancel out with +2y in the second equation.
- You have the equations 6x – 2y = 6 and -x + 2y = 4 .
- Add the two left sides: 6x – 2y – x + 2y = ?
- Add the two right sides: 6x – 2y – x + 2y = 6 + 4 .
- You have 6x – 2y – x + 2y = 6 + 4 .
- Group the variables x and y together: 6x – x – 2y + 2y = 6 + 4 .
- Simplify the equation we have: 5x = 10
- Solve for x: (5x)/5 = 10/5 , so x = 2 .
- You know that x = 2 , and one of the two original equations is 3x – y = 3 .
- Replace 2 in x: 3(2) – y = 3 .
- Solve for y in this equation: 6 – y = 3
- 6 – y + y = 3 + y , so 6 = 3 + y
- 3 = y
- If the equation adds up with no variables and is not true (like 2 = 7) then the system of two equations has no solution . (If you graph the two equations, you’ll see that they’re parallel and never intersect.)
- If the equation adds up with no variables but is true (like 0 = 0) then there are infinitely many solutions . These two equations are actually exactly the same. (If you plot the two equations, you’ll see that they are a single line.)
Plot the equation
- The basic idea is to graph the two equations and find their intersection. The x and y values at this point are the values of x and y in the system of equations.
- The first equation is 2x + y = 5 . Transform it to y = -2x + 5 .
- The second equation is -3x + 6y = 0 . Transform it to 6y = 3x + 0 , then reduce it to y = ½x + 0 .
- If both equations are identical then the entire line will be the “intersection”. Writing is infinitely many solutions .
- If you don’t have graph paper, use a ruler to make sure the numbers are evenly spaced.
- If you are using large numbers or decimals, you will need to scale the graph differently. (For example 10, 20, 30 or 0.1; 0.2; 0.3 but not 1, 2, 3).
- In our original example, the line ( y = -2x + 5 ) intersects the y axis at 5 . The line ( y = ½x + 0 ) intersects the y axis at 0 . (These are the points (0.5) and (0,0) on the graph.)
- If possible, you should use a different color pen to draw two lines.
- In the above example, the line y = -2x + 5 has a slope of -2 . At x = 1, the line moves down 2 units from the point with coordinates x = 0. Draw a line segment between the two points (0.5) and (1,3).
- The line y = ½x + 0 has a slope of ½ . At x = 1, the line moves up ½ unit from the point with coordinates x = 0. Draw a line segment between the two points (0,0) and (1,½).
- If two lines have the same slope , they never intersect, so the system of equations has no solution. Writing is futile .
- If two lines are moving towards each other then you continue to define more points in that direction.
- If the two lines are moving away from each other, locate the points of the graph in the opposite direction, starting with x = -1.
- If the two lines are very far apart, then you determine the very far points of the graph, for example x = 10.
Advice
- You can check your work by replacing your answers with the original equations. If the equations are correct (eg 3 = 3) then the answer is correct.
- In the suppression method, you will sometimes have to multiply an equation by a negative number to cancel out a variable.
Warning
- You cannot use these methods if the equation has a variable that contains an exponent, such as x 2 . For more information on equations of this form, look up the literature on factoring quadratic equations. [5] X Research Sources
wikiHow is a “wiki” site, which means that many of the articles here are written by multiple authors. To create this article, 50 people, some of whom are anonymous, have edited and improved the article over time.
This article has been viewed 9,870 times.
For “system of equations” you will have to solve two or more equations at the same time. When these equations have two different variables, such as x and y or a and b, you may not know how to solve them at first. Luckily, all you need is basic algebra skills to solve a system of equations (sometimes a little knowledge of fractions is required). If you are in the habit of learning visually, or if your teacher requires it, you should learn how to graph equations. Graphing can be a useful way to “see what’s going on” or to check results, but this is slower than other solutions and doesn’t work for every system of equations.
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