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Systems of Equations
Systems of Equations in Two Variables
Algebra Textbooks Boundless Algebra Systems of Equations Systems of Equations in Two Variables
Algebra Textbooks Boundless Algebra Systems of Equations
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Algebra Textbooks
Algebra
Concept Version 11
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The Substitution Method

The substitution method is a way of solving a system of equations by expressing the equations in terms of only one variable.

Learning Objective

  • Solve systems of equations in two variables using substitution


Key Points

    • A system of equations is a set of equations that can be solved using a particular set of values.
    • The substitution method works by expressing one of the variables in terms of another, then substituting it back into the original equation and simplifying it.
    • It is very important to check your work once you have found a set of values for the variables. Do this by substituting the values you found back into the original equations.
    • The solution to the system of equations can be written as an ordered pair (x,y).

Terms

  • system of equations

    A set of equations with multiple variables which can be solved using a specific set of values.

  • substitution method

    Method of solving a system of equations by putting the equation in terms of only one variable


Full Text

The substitution method for solving systems of equations is a way to simplify the system of equations by expressing one variable in terms of another, thus removing one variable from an equation. When the resulting simplified equation has only one variable to work with, the equation becomes solvable. 

The substitution method consists of the following steps:

  1. In the first equation, solve for one of the variables in terms of the others.
  2. Substitute this expression into the remaining equations.
  3. Continue until you have reduced the system to a single linear equation.
  4. Solve this equation, and then back-substitute until the solution is found.

Solving with the Substitution Method

Let's practice this by solving the following system of equations:

$x-y=-1$

$x+2y=-4$

We begin by solving the first equation so we can express x in terms of y. 

$\begin{aligned} \displaystyle x-y&=-1 \\x&=y-1 \end{aligned}$

Next, we will substitute our new definition of x into the second equation:

$\displaystyle \begin{aligned} x+2y&=-4 \\(y-1)+2y&=-4 \end{aligned}$

Note that now this equation only has one variable (y). We can then simplify this equation and solve for y:

$\displaystyle \begin{aligned} (y-1)+2y&=-4 \\3y-1&=-4 \\3y&=-3 \\y&=-1 \end{aligned}$

Now that we know the value of y, we can use it to find the value of the other variable, x. To do this, substitute the value of y into the first equation and solve for x.

$\displaystyle \begin{aligned} x-y&=-1 \\x-(-1)&=-1 \\x+1&=-1 \\x&=-1-1 \\x&=-2 \end{aligned}$

Thus, the solution to the system is: $(-2, -1)$, which is the point where the two functions graphically intersect. Check the solution by substituting the values into one of the equations.

$\displaystyle \begin{aligned} x-y&=-1 \\(-2)-(-1)&=-1 \\-2+1&=-1 \\-1&=-1 \end{aligned} $

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