Math Help for Section 8.1, Page 378

Systems
of Linear Equations

Until this chapter, most problems have involved just one equation in
either one or
two variables. However, many problems in science, business, health
services, and
government involve two or more equations in two or more variables. The
finance problem described on page 354 is one example. 

Example
1: Tip
Checking a solution of a system of equations works the same way as
checking a solution of a single equation (introduced in the Math Help
for
section 2.4, page 86). However, you have to substitute more than one
variable into the equations and there is more than one equation to
check. When you substitute coordinates into two equations and the
substitution results in a true statement in both equations,
then the coordinate point is a solution of the system of equations. When
the substitution results in
a false statement in one
or both
equations, then the coordinate point is not a solution. For instance,
check whether $(-2,5)$ is a solution of the system of
equations in Example 1.

In the first equation, $3x+2y=4$, substitute $-2$ for x and 5 for y.

  
 $3({\color{red}-2})+2({\color{red}5})\overset{?}{=}4\quad\longrightarrow\quad
-6+10=4\quad\quad\eqalign{&\small\color{red}\text{Solution checks}
\cr &\small\color{red}\text{in 1st equation.}}$

In the second equation, $-x+3y=-5$, substitute $-2$ for x and 5 for y.

  
 $-({\color{red}-2})+3({\color{red}5})\overset{?}{=}-5\quad\longrightarrow\quad
-2+15\ne
-5\quad\quad\eqalign{&\small\color{red}\text{Solution does not}
\cr &\small\color{red}\text{check in 2nd equation.}}$

Because the solution $(-2,5)$ does not check in both
equations, you can conclude that it is not a solution of the original
system of equations. Also, you can see that $(-2,5)$ is not the point of intersection in
the graph.