Systems of equations are two or more equations with two or more variables that have to be solved simultaneously. A solution to a system of equations satisfies all the equations in the system at the same time.
Here, we will look at various examples of systems of two equations with two unknowns. We will use the elimination method and the substitution method to solve the systems. We’ll start with a quick review of these methods and then look at several worked examples.
Systems of equations and solutions
Systems of equations are sets of two or more equations, which have to be solved simultaneously. In this article, we will focus entirely on systems of two equations with two unknowns and we will solve exercises only with these systems.
Solution to a system
For an ordered pair to be a solution to a system of equations, it has to satisfy both equations at the same time. If an ordered pair is a solution to one equation but not the other, then it is not a solution to the system of equations.
A system that has at least one solution is called a consistent system.
A system that has no solutions is called an inconsistent system.
There are three possible results that we can find in the systems of equations:
1. One solution: this happens when the lines that represent the equations intersect at a single point.
2. No solution: this happens when the lines that represent the equations are parallel.
3. Infinite solutions: this happens when the line of one equation overlaps the line of the other equation.
Solving systems of equations with the substitution method
Follow these steps to solve the systems of equations problems with the substitution method:
Step 1: Simplify if possible.
Remove parentheses, combine like terms, and eliminate fractions.
Step 2: Solve an equation for one variable.
It doesn’t matter which equation we choose or for which variable we solve.
Step 3: Substitute the expression from step 2 into the other equation.
By doing this, we will obtain an equation with only one unknown.
Step 4: Solve for the variable we have remaining.
If you need help with this, you can look at our guide on how to solve equations with one unknown.
Step 5: Solve for the other variable.
We substitute the value obtained in step 4 into any equation and solve.
Step 6: Check the solution in both equations.
We plug the values into the equations and check the result.
EXAMPLE 1
Solve with the substitution method: $latex \left\{ {\begin{array}{*{20}{c}} {5x-y=2} \\ {y=4x-1} \end{array}} \right.$.
Solution
Step 1: Simplify if possible.
Both equations are already simplified.
Step 2: Solve an equation for one variable.
The second equation is already solved for the variable y:
$latex y=4x-1$
Step 3: Substitute the expression from step 2 into the other equation.
We substitute the expression $latex y=4x-1$ in the first equation:
$latex 5x-y=2$
$latex 5x-(4x-1)=2$
$latex 5x-4x+1=2$
Step 4: Solve for the remaining variable:
$latex 5x-4x+1=2$
$latex x=1$
Step 5: Solve for the second variable.
We substitute $latex x=1$ in the second equation:
$latex y=4x-1$
$latex y=4(1)-1$
$latex y=3$
Step 6: Check the solution in both equations.
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EXAMPLE 2
Solve with the substitution method: $latex \left\{ {\begin{array}{*{20}{c}} {3x+y=11} \\ {2x-y=-1} \end{array}} \right.$
Solution
Step 1: Simplify if possible.
Both equations are already simplified.
Step 2: Solve an equation for one variable.
We solve the second equation for y:
$latex 2x-y=-1$
$latex y=2x+1$
Step 3: Substitute the expression from step 2 into the other equation.
We substitute the expression $latex y=2x+1$ n the first equation:
$latex 3x+y=11$
$latex 3x+2x+1=11$
Step 4: Solve for the remaining variable:
$latex 3x+2x+1=11$
$latex 5x=10$
$latex x=2$
Step 5: Solve for the second variable.
We substitute$latex x=2$ in the second equation:
$latex 2x-y=-1$
$latex 2(2)-y=-1$
$latex 4-y=-1$
$latex y=5$
Step 6: Check the solution in both equations.
EXAMPLE 3
Solve with the substitution method: $latex \left\{ {\begin{array}{*{20}{c}} {4x-2y=15} \\ {3x+2y=25} \end{array}} \right.$
Solution
Step 1: Simplify if possible.
Both equations are already simplified.
Step 2: Solve an equation for one variable.
We solve the second equation for y:
$latex 3x+2y=25$
$latex 2y=25-3x$
$latex y=\frac{{25}}{2}-\frac{3}{2}x$
Step 3: Substitute the expression from step 2 into the other equation.
We substitute the expression $latex y=\frac{{25}}{2}-\frac{3}{2}x$ in the first equation:
$latex 4x-2y=15$
$latex 4x-2(\frac{{25}}{2}-\frac{3}{2}x)=15$
$latex 4x-25+3x=15$
Step 4: Solve for the remaining variable:
$latex 4x-25+3x=15$
$latex 7x=35$
$latex x=5$
Step 5: Solve for the second variable.
We substitute $latex x=5$ in the second equation:
$latex 3x+2y=25$
$latex 3(5)+2y=25$
$latex 2y=25-15$
$latex y=5$
Step 6: Check the solution in both equations.
EXAMPLE 4
Solve with the substitution method: $latex \left\{ {\begin{array}{*{20}{c}} {6x+3y=-9} \\ {2x-2y=-12} \end{array}} \right.$
Solution
Step 1: Simplify if possible.
We can simplify the first equation by dividing it by 3: $latex 2x+y=-3$.
We can simplify the second equation by dividing it by 2: $latex x-y=-6$.
Step 2: Solve an equation for one variable.
We solve the first equation for y:
$latex 2x+y=-3$
$latex y=-3-2x$
Step 3: Substitute the expression from step 2 into the other equation.
We substitute the expression $latex y=-3-2x$ in the second equation:
$latex x-y=-6$
$latex x-(-3-2x)=-6$
$latex x+3+2x=-6$
Step 4: Solve for the remaining variable:
$latex x+3+2x=-6$
$latex 3x=-9$
$latex x=-3$
Step 5: Solve for the second variable.
We substitute $latex x=-3$ in the second equation:
$latex x-y=-6$
$latex -3-y=-6$
$latex -y=-3$
$latex y=3$
Step 6: Check the solution in both equations.
Try solving the following practice problems
Solve systems with the elimination method
Follow these steps to solve the systems of equations problems with the substitution method:
Step 1: Simplify if possible and put the equations in the form Ax + By = C.
Step 2: Multiply one or both equations by a number to create opposite coefficients for either x or y.
We are going to add the equations and we have to make one of the variables disappear. We can achieve this with opposite coefficients since their sum is 0.
For example, if we have $latex 3x$ in the first equation and $latex 2x$ in the second equation, we can multiply the first by -2 and the second by 3, thus we get -6 in the first and 6 in the second.
Step 3: Add the equations.
One of the variables will be eliminated and we will obtain an equation with only one unknown.
Step 4: Solve for the remaining variable.
If you need help with this, you can look at our guide on how to solve equations with one unknown.
Step 5: Solve for the second variable.
Substitute the value you found in step 4 into either of the two equations and solve for the other variable.
Step 6: Check the solution in both equations.
EXAMPLE 1
Solve with elimination method: $latex \left\{ {\begin{array}{*{20}{c}} {3x+3y=15} \\ {-3x+y=-3} \end{array}} \right.$
Solution
Step 1: Simplify if possible and write in the form Ax+By=C.
Both equations are already simplified and in the form Ax+By=C.
Step 2: Multiply one or both equations by a number that will create opposite coefficients for x or for y.
We already have opposite coefficients in the variable x.
Step 3: Add the equations.
$latex 3x+3y=15$
$latex + \hspace{1cm} -3x+y=-3$
___________________
$latex 4y=12$
Step 4: Solve for the remaining variable:
$latex 4y=12$
$latex y=3$
Step 5: Solve for the second variable.
We substitute $latex y=3$ in the first equation:
$latex 3x+3y=15$
$latex 3x+3(3)=15$
$latex 3x+9=15$
$latex 3x=6$
$latex x=2$
Step 6: Check the solution in both equations.
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EXAMPLE 2
Solve with elimination method: $latex \left\{ {\begin{array}{*{20}{c}} {-2x+y=9} \\ {-3x-2y=3} \end{array}} \right.$
Solution
Step 1: Simplify if possible and write in the form Ax+By=C.
Both equations are already simplified and in the form Ax+By=C.
Step 2: Multiply one or both equations by a number that will create opposite coefficients for x or for y.
We multiply the first equation by 2 to eliminate the y:
$latex \left\{ {\begin{array}{*{20}{c}} {-4x+2y=18} \\ {-3x-2y=3} \end{array}} \right.$
Step 3: Add the equations.
$latex -4x+2y=18$
$latex + \hspace{1cm} -3x-2y=3$
___________________
$latex -7x=21$
Step 4: Solve for the remaining variable:
$latex -7x=21$
$latex x=-3$
Step 5: Solve for the second variable.
We substitute $latex x=-3$ in the first equation:
$latex -2x+y=9$
$latex -2(-3)+y=9$
$latex 6+y=9$
$latex y=3$
Step 6: Check the solution in both equations.
EXAMPLE 3
Solve with the elimination method: $latex \left\{ {\begin{array}{*{20}{c}} {x=4y+5} \\ {2(x+2)+3y=3} \end{array}} \right.$
Solution
Step 1: Simplify if possible and place in the form Ax+By=C:
$latex \left\{ {\begin{array}{*{20}{c}} {x-4y=5} \\ {2x+3y=1} \end{array}} \right.$
Step 2: Multiply one or both equations by a number that will create opposite coefficients for x or for y.
We multiply the first equation by -2 to eliminate the x:
$latex \left\{ {\begin{array}{*{20}{c}} {-2x+8y=-10} \\ {2x+3y=1} \end{array}} \right.$
Step 3: Add the equations.
$latex -2x+8y=10$
$latex + \hspace{1cm} 2x+3y=1$
___________________
$latex 11x=11$
Step 4: Solve for the remaining variable:
$latex 11x=11$
$latex x=1$
Step 5: Solve for the second variable.
We substitute$latex x=1$ in the first equation:
$latex x=4y+5$
$latex 1=4y+5$
$latex -4y=4$
$latex y=-1$
Step 6: Check the solution in both equations.
EXAMPLE 4
Solve with elimination method: $latex \left\{ {\begin{array}{*{20}{c}} {2x+2y=20} \\ {-3x+3y=0} \end{array}} \right.$
Solution
Step 1: Simplify if possible and write in the form Ax+By=C.
We can simplify both equations: we divide the first by 2 and the second by 3:
$latex \left\{ {\begin{array}{*{20}{c}} {x+y=10} \\ {-x+y=0} \end{array}} \right.$
Step 2: Multiply one or both equations by a number that will create opposite coefficients for x or for y.
We already have opposite coefficients in the x:
Step 3: Add the equations.
$latex x+y=10$
$latex + \hspace{1cm} -x+y=0$
___________________
$latex 2y=10$
Step 4: Solve for the remaining variable:
$latex 2y=10$
$latex y=5$
Step 5: Solve for the second variable.
We substitute$latex y=5$ in the second equation:
$latex -x+y=0$
$latex -x+5=0$
$latex x=5$
Step 6: Check the solution in both equations.
Try solving the following practice problems
See also
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