EDGE Math 8

Example 2

Solve the system of linear equations using the substitution method. Tell what type of system of equations it is.

{\begin{array}{l} 2 x - y = 3 \\ 4 x - 2 y = 6 \end{array}

Solution

Use the first equation to solve for y in terms of x.

\begin{array}{l} - y = 3 - 2 x \\ y = - 3 + 2 x \end{array}

Substitute $- 3 + 2 x$ for y in the second equation of the system.

$4 x - 2 (- 3 + 2 x)$	$= 6$
$4 x + 6 - 4 x$	$= 6$
$4 x - 4 x$	$= 6 - 6$
$0$	$= 0$

The last equation $(0 = 0)$ indicates that the value of x is any real number.
Since $y = - 3 + 2 x,$ the solution set consists of all ordered pairs that satisfy the equation $y = - 3 + 2 x .$ These ordered pairs are in the form $(x, - 3 + 2 x),$ where x is any real number. Examples of these ordered pairs are $(0, - 3)$ and $(1, - 1) .$

In set notation, the solution set is given by

{(x, y) | y = - 3 + 2 x} .

The system has an infinite number of solutions. It is consistent and dependent.

Substitution Method