Systems of Equations in Two Variables
(Adapted from UTLC)
A system of equations in two unknowns can be used to solve an equation which has two unknowns.
Consider the following question:
If you graph two equations on an x-y plane, in how many ways can the two lines intersect? In other words, what points (x- and y-values) will be the same in each case?
Figure 1: Graphic Solutions
Intersection at One Point |
No Intersection |
Intersection at All Points |
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To solve a system of equations in two unknowns we need to find all of the intersecting points. In other words, we need to find all of the x- and y-values that make both equations true.
What would the answers look like for each case in Figure 1 above if the system is solved algebraically? Knowing what the solutions look like graphically, how do you solve the system algebraically? There are two methods:
- Substitution
- Elimination by Addition
Example 1
Solve the following system of equations graphically and by both algebraic methods.
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Graphically |
Substitution |
Elimination |
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- Choose one of the equations and solve for one of the variables. In this example we will choose equation (a) and solve for y by adding –x to both sides of the equation.
- Put this value of y into the other equation. In this example put the value for y into equation b.
This gives us one equation in one unknown so we can solve for x.
- Put this x-value back into either equation to solve for y.
- The point where the two lines intersect is ( 2 , 2).
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- First, arrange both equations so that the terms with the same variables line up.
- Now multiply equation (a) by some number that will make the coefficients of either x or y additive inverses. In our example multiply equation (a) by -2.
- Put this value for x into either equation to solve for y.
- As in the Substitution method, we find that the two graphs intersect at ( 2 , 2).
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Example 1 leads us to the conclusion that a system that intersects at one point has one solution which can be written in the form ( x , y ).
What about parallel lines? What does the algebraic solution to a system of parallel lines look like?
Example 2
Solve the following system of equations graphically and by both algebraic methods.
Note; The two equations in Example 2 have the same slope but different y-intercepts. Therefore, these lines are parallel.
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Graphically |
Substitution |
Elimination |
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- Substitute the value of y from equation (a) into equation (b).
- Since the x-terms are the same on both sides of the equation they cancel out, leaving a false statement. Therefore, since
 , the system does not have any solutions. Because the two lines are parallel they will never intersect. |
- Multiply equation (b) by -1.
- Add this new equation to equation (a).
- Once again we get a false statement since
 . Therefore, the lines do not intersect and the system has no solutions. |
We conclude from Example 2 that a system that does not intersect (the lines are parallel) has no solutions. When you solve such a system algebraically the variable terms add up to zero and you will always get a false statement. The answer is “no solutions” or 
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What about two lines that have the same slope as well as the same y-intercept? What does the algebraic solution to this type of system look like?
Example 3
Solve the following system of equations graphically and by both algebraic methods.
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Graphically |
Substitution |
Elimination |
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- Divide equation (b) by 6.
- Substitute this value for y in equation (b) into equation (a). Then solve for x.
 
- This implies that these two equations produce the same line. Further, there are an infinite amount of solutions. Any point you choose on line (a) will be a point on line (b)
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- Rewrite equation (b) so that the x- and y-terms are on the left side of the equation.
- Add this new form of equation (b) to equation (a).
- Multiply equation (a) by 3 and add to equation (b).
- As in the substitution method, this shows that these two equations represent the same line and there are an infinite amount of solutions.
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From Example 3 we can conclude that two lines that are the same intersect at all points and the answer is “infinitely many solutions.”
Figure 2: Summary
Intersection at One Point
Intersecting Lines |
No Intersection
Parallel Lines |
Intersection at All Points
Same Lines |
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- Lines have different slopes
- Lines may have the same y-intercept or it may be different
- Both algebraic solutions will give you an x-value and a y-value
- The answer will be in the form ( x, y )
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- Lines have the same slope
- Lines have different y-intercepts
- Both algebraic solutions will give you a false statement
- The answer will be “no solutions” or
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- Lines have the same slope
- Lines have the same y-intercept
- Both algebraic solutions will give you a true statement
- The answer will be “infinitely many solutions”
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