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1. Analyzing Functions
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United States of AmericaTennessee
Algebra 2

1.06 Systems of linear equations

Lesson
Practice
Lesson

Concept summary

There are many different methods for solving systems of linear equations. Some of which include: graphing method, substitution method, and the elimination method.

Graphing method

A method of solving a system of equations by graphing the equations and finding the point of intersection.

This method is best used when graphs are presented within a question and the intersection point is easily identifiable.

Substitution method

A method of solving a system of equations by replacing a variable in one equation with an equivalent expression from another equation.

This method is best used when at least one equation in a system of equations can be solved for one variable.

Elimination method

A method of solving a system of equations by adding or subtracting the equations until only one variable remains.

This method is best used if at least one pair of variables in a system of equations can be made to have the same coefficient, or already have the same coefficient.

For all systems of equations, a solution in a given context is said to be viable if the solution makes sense in the context, and non-viable if it does not make sense within the context, even if it would otherwise be algebraically valid.

We can also apply these same solving methods to systems of equations with more than 2 equations. For example, we may need to solve a system with 3 equations and 3 unknown variables like this: \begin{cases} x + y + z = 20\\4x - 2y + 3z = 14\\3 \left(x + y\right) = z \end{cases}

The rule of thumb when it comes to solving systems of equations is that we need at least the same number of equations as we have variables. This means for 3 unknown variables, we need at least 3 equations.

Worked examples

Example 1

Consider the following system of equations:\begin{cases} y = 2 x + 8 \\ x + y = 5 \end{cases}

a

Sketch the two equations on a coordinate plane.

Approach

We can rearrange the second equation so that it is also in slope-intercept form to make graphing easier.

\displaystyle x+y\displaystyle =\displaystyle 5Second equation
\displaystyle y\displaystyle =\displaystyle -x + 5Subtract x from both sides

Now both of the equations are in slope-intercept form and we can graph them on a coordinate plane.

Solution

-4
-3
-2
-1
1
2
3
4
x
-1
1
2
3
4
5
6
7
8
y
b

Solve the system of equations.

Approach

Because we have the graphs from part (a) and the intersection point is easily identifiable, we can use the graphing method to solve this system of equations.

Solution

To solve a system of equations by graphing, we need to identify the intersection point of the graphs. The intersection point is \left(-1, 6 \right).

Example 2

Solve the following system of equations with an efficient method:\begin{cases}5x+3y\ =31 \\ 16-x=2y \end{cases}

Approach

Because we can solve for one of the variables, specifically x in the second equation, we will use the substitution method.

We can number our equations to make the system easier to work with.

1\displaystyle 5x + 3 y\displaystyle =\displaystyle 31
2\displaystyle 16 - x\displaystyle =\displaystyle 2y

Solution

We'll start by isolating x in equation 2.

\displaystyle 16 - x\displaystyle =\displaystyle 2yEquation 2
\displaystyle 16\displaystyle =\displaystyle 2y+xAdd x to both sides
\displaystyle 16-2y\displaystyle =\displaystyle xSubtract 2y from both sides

We've isolated x which is equal to 16-2y.

We can now substitute this value back into equation 1 to solve for y.

\displaystyle 5\left(16-2y\right) +3y\displaystyle =\displaystyle 31Substitute x=16-2y into Equation 1
\displaystyle 80 - 10y + 3y \displaystyle =\displaystyle 31Distributive property
\displaystyle 80 - 7y\displaystyle =\displaystyle 31Combine y terms
\displaystyle -7y\displaystyle =\displaystyle -49Subtract 80 from both sides
\displaystyle y\displaystyle =\displaystyle 7Divide both sides by -7

Therefore y=7. We can substitute this back into either equation in order to solve for x.

\displaystyle 5x+3y\displaystyle =\displaystyle 31Equation 1
\displaystyle 5x+3(7)\displaystyle =\displaystyle 31Substitute y=7 into Equation 2
\displaystyle 5x+21\displaystyle =\displaystyle 31Simplify the product
\displaystyle 5x\displaystyle =\displaystyle 10Subtract 21 from both sides
\displaystyle x\displaystyle =\displaystyle 2Divide both sides by 5

Therefore x=2. We can put it all together and say that the solution to our system of equations is \left(2, 7 \right).

Reflection

This system can also be solved using the elimination method if we rearrange the terms so that they appear in the same order.

\displaystyle 16-x\displaystyle =\displaystyle 2yEquation 2
\displaystyle 16\displaystyle =\displaystyle 2y+xAdd x to both sides
\displaystyle 2y+x\displaystyle =\displaystyle 16Symmetric property
\displaystyle x+2y\displaystyle =\displaystyle 16Commutative property of addition

The system is now:

\displaystyle 5x + 3 y\displaystyle =\displaystyle 31
\displaystyle x+2y\displaystyle =\displaystyle 16

Since the coefficients are not the same we can need to multiply Equation 2 by 5.

\displaystyle 5x + 3 y\displaystyle =\displaystyle 31
\displaystyle 5x+10y\displaystyle =\displaystyle 80

Now that the coefficients of both x terms are the same we can solve for y by subtracting Equation 2 from Equation 1:

\displaystyle (5x-5x)+(3y-10y)\displaystyle =\displaystyle (31-80)Equation 2 subtracted from Equation 1
\displaystyle -7y\displaystyle =\displaystyle -49Combine like terms
\displaystyle y\displaystyle =\displaystyle 7Divide both sides by -7

From here we can solve for the value of x in the same way as we did before.

Example 3

Use technology to solve the following system of equations.\begin{cases} 3x-2y+z=15 \\ x+ 8y - 2z = -11 \\ \frac{1}{2}x-4y-\frac{1}{2}z=1 \end{cases}

Approach

We can use technology to help us solve this system of equations. We will plot the 3 equations with a 3D calculator and find the intersection point of the equations.

To help identify and keep track of the equations, we can label them as equations 1-3:

1\displaystyle 3x-2y+z\displaystyle =\displaystyle 15
2\displaystyle x+ 8y - 2z\displaystyle =\displaystyle -11
3\displaystyle \frac{1}{2}x-4y-\frac{1}{2}z\displaystyle =\displaystyle 1

Solution

Open the calculator tool and type the first function into the calculator's input. You can repeat this process for the remaining 2 equations.

The 3D graphing calculator tool with 3 planes graphed. Speak to your teacher for more info.

We now need to find the intersection point. We can do this by first finding the line of intersection of equations 1 and 2, followed by the line of intersection of equations 2 and 3. Afterwards we can find the intersection of those two lines which will be our common point of intersection for all 3 equations.

You can find the intersection line of two equations by going to "Tools", and clicking "Intersect Two Surfaces".

The 3D calculator tool with 3 planes graphed showing the basic tools menu. Speak to your teacher for more info.

We will repeat this twice, once for equations 1 and 2, then equations 2 and 3. The results will look like this:

The 3D calculator tool with 3 planes graphed and a line graphed at the intersection of each pair of planes. Speak to your teacher for more info.

The final step is to find the intersection point between our two lines. We can do this by going to tools and clicking "More", find the "Points" section and click "Intersect".

The 3D calculator tool with 3 planes graphed and a line graphed at the intersection of each pair of planes and the basic tools menu shown. Speak to your teacher for more info.

Once you intersect the two lines, you can go back to "Algebra" and you will see the resulting point of intersection.

The 3D calculator tool with 3 planes graphed and a line graphed at the intersection of each pair of planes. A point is graphed at the intersection of the lines. Speak to your teacher for more info.

As you can see the intersection of our two lines is at the point \left(3, -\dfrac{1}{2}, 5 \right). Therfore the solution to our system of equations is x=3, y=-\dfrac{1}{2}, and z=5.

Outcomes

A2.N.Q.A.1

Use units as a way to understand real-world problems.*

A2.A.REI.B.3

Write and solve a system of linear equations in a real-world context. *

A2.MP1

Make sense of problems and persevere in solving them.

A2.MP2

Reason abstractly and quantitatively.

A2.MP3

Construct viable arguments and critique the reasoning of others.

A2.MP4

Model with mathematics.

A2.MP5

Use appropriate tools strategically.

A2.MP6

Attend to precision.

A2.MP7

Look for and make use of structure.

A2.MP8

Look for and express regularity in repeated reasoning.

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