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Solving a linear system in two variables by graphing works well when the solution consists of integer values, but if our solution contains decimals or fractions, it is not the most precise method. We will consider two more methods of solving a system of linear equations that are more precise than graphing. One such method is solving a system of equations by the substitution method , in which we solve one of the equations for one variable and then substitute the result into the second equation to solve for the second variable. Recall that we can solve for only one variable at a time, which is the reason the substitution method is both valuable and practical.
Given a system of two equations in two variables, solve using the substitution method.
Solve the following system of equations by substitution.
First, we will solve the first equation for $\text{\hspace{0.17em}}y.$
Now we can substitute the expression $\text{\hspace{0.17em}}x\mathrm{-5}\text{\hspace{0.17em}}$ for $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ in the second equation.
Now, we substitute $\text{\hspace{0.17em}}x=8\text{\hspace{0.17em}}$ into the first equation and solve for $\text{\hspace{0.17em}}y.$
Our solution is $\text{\hspace{0.17em}}\left(8,3\right).$
Check the solution by substituting $\text{\hspace{0.17em}}\left(8,3\right)\text{\hspace{0.17em}}$ into both equations.
Solve the following system of equations by substitution.
$\left(\mathrm{-2},\mathrm{-5}\right)$
Can the substitution method be used to solve any linear system in two variables?
Yes, but the method works best if one of the equations contains a coefficient of 1 or –1 so that we do not have to deal with fractions.
A third method of solving systems of linear equations is the addition method . In this method, we add two terms with the same variable, but opposite coefficients, so that the sum is zero. Of course, not all systems are set up with the two terms of one variable having opposite coefficients. Often we must adjust one or both of the equations by multiplication so that one variable will be eliminated by addition.
Given a system of equations, solve using the addition method.
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