17Calculus Precalculus - Use Substitution to Solve Systems of Linear Equations

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Solve by Substitution

Another technique to solve systems of equations is substitution. Just like graphing, this technique is limited to two equations with two unknowns since higher order systems become quite complicated very quickly. Also this technique will often produce fractions very quickly, making the algebra more difficult. However, this is a handy technique to learn since you will use it later on with more effect.
Here is a good video showing this technique using an example. He doesn't check his answer in this video, which is not something you want to skip. Also, notice that his answer contains fractions, so graphing would not be the way to solve this problem.

MathOnPoint - Solving Linear Systems Substitution Method [7min-58secs]

video by MathOnPoint

Okay, this technique is pretty easy to understand, so we are going to let you jump right into the practice problems.

Practice

Unless otherwise instructed, solve these linear systems using substitution.

Basic

$$2x+4y=4$$
$$y=x-2$$

Problem Statement

$$2x+4y=4$$
$$y=x-2$$

Solution

PatrickJMT - 1710 video solution

video by PatrickJMT

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$$x+3y=6$$
$$2x+6y=-12$$

Problem Statement

$$x+3y=6$$
$$2x+6y=-12$$

Solution

PatrickJMT - 1711 video solution

video by PatrickJMT

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$$2x-3y=6$$
$$4x-6y=12$$

Problem Statement

$$2x-3y=6$$
$$4x-6y=12$$

Solution

PatrickJMT - 1712 video solution

video by PatrickJMT

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$$x+2y=9$$
$$3x+5y=20$$

Problem Statement

$$x+2y=9$$
$$3x+5y=20$$

Solution

Khan Academy - 1713 video solution

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$$y=3x$$
$$2x+y=-10$$

Problem Statement

$$y=3x$$
$$2x+y=-10$$

Solution

MIP4U - 1715 video solution

video by MIP4U

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$$x-2y=1$$
$$3x-6y=-18$$

Problem Statement

$$x-2y=1$$
$$3x-6y=-18$$

Solution

MIP4U - 1716 video solution

video by MIP4U

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$$x-2y=7$$
$$2y-3x=-1$$

Problem Statement

$$x-2y=7$$
$$2y-3x=-1$$

Solution

MIP4U - 1717 video solution

video by MIP4U

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$$2x-y=-5$$
$$4x-2y=-10$$

Problem Statement

$$2x-y=-5$$
$$4x-2y=-10$$

Solution

MIP4U - 1718 video solution

video by MIP4U

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$$x+3y=12; 2x+y=6$$

Problem Statement

$$x+3y=12; 2x+y=6$$

Solution

MathOnPoint - 2129 video solution

video by MathOnPoint

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Intermediate

Here is a little more difficult problem that consists of 3 equations and 3 unknowns. Try substitution here as well.

$$7x-4y-2z=7; 8x-2y-8z=1; x+2y-6z=-6$$

Problem Statement

$$7x-4y-2z=7; 8x-2y-8z=1; x+2y-6z=-6$$

Solution

PatrickJMT - 1714 video solution

video by PatrickJMT

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