Solve the system of equations by substitution

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Systems of Linear equations:

A system of linear equations is just a set of two or more linear equations.

In two variables ( x and y ) , the graph of a system of two equations is a pair of lines in the plane.

There are three possibilities:

The lines intersect at zero points. (The lines are parallel.)
The lines intersect at exactly one point. (Most cases.)
The lines intersect at infinitely many points. (The two equations represent the same line.)

How to Solve a System Using The Substitution Method

Step 1 : First, solve one linear equation for y in terms of x .
Step 2 : Then substitute that expression for y in the other linear equation. You'll get an equation in x .
Step 3 : Solve this, and you have the x -coordinate of the intersection.
Step 4 : Then plug in x to either equation to find the corresponding y -coordinate.

Note 1 : If it's easier, you can start by solving an equation for x in terms of y , also – same difference!

Example:

Solve the system { 3 x + 2 y = 16 7 x + y = 19

Solve the second equation for y .

y = 19 − 7 x

Substitute 19 − 7 x for y in the first equation and solve for x .

3 x + 2 ( 19 − 7 x ) = 16 3 x + 38 − 14 x = 16 − 11 x = − 22 x = 2

Substitute 2 for x in y = 19 − 7 x and solve for y .

y = 19 − 7 ( 2 ) y = 5

Note 2 : If the lines are parallel, your x -terms will cancel in step 2 , and you will get an impossible equation, something like 0 = 3 .

Note 3 : If the two equations represent the same line, everything will cancel in step 2 , and you will get a redundant equation, 0 = 0 .

What is the most useful technique for solving a system of equations?

Jenn, Founder Calcworkshop®, 15+ Years Experience (Licensed & Certified Teacher)

The Substitution Method!

Why?

Because it is used in such topics as nonlinear systems, linear algebra, computer programming, and so much more.

And the greatest thing about solving systems by substitution is that it’s easy to use!

The method of substitution involves three steps:

Solve one equation for one of the variables.
Substitute (plug-in) this expression into the other equation and solve.
Resubstitute the value into the original equation to find the corresponding variable.

Now at first glance, this may seem complicated, but I’ve got some helpful tricks for keeping things straight. In fact, we’re going to make a sort of circular circuit that helps to provide organization and efficiency to our method.

Using the Substitution Method to Solve

Remember, our goal when solving any system is to find the point of intersection. As we saw in our lesson titled the graphing method, we saw that some systems do not have solutions because they don’t intersect, and others coincide, which provides infinitely many solutions.

So when we solve systems by substitution, we will need to be on the lookout for these types of scenarios. If they are parallel and don’t intersect, then we are going to end up with an invalid answer, or as Purple Math calls it, a “garbage” result.

Together we will look at 11 examples of solving linear systems using the substitution method, and learn how to employ this technique for systems of two, three and even four equations.