Math Help for Section 3.1, Page 104
Solving
Linear Equations in Nonstandard Form
The definition of linear equation contains the phrase “that can be
written in the standard form $ax + b = 0.$” This suggests
that
some linear equations may come in nonstandard or disguised form.
A common type of linear
equation is one
in which the variable terms are not combined into one term. In such
cases, you can begin the solution by combining like terms.
Note how
this is done in Examples 3 and 4.
Example
3: Tip
Note that the variable in the equation does not always have to be x. You
can use any letter. Also, you can isolate the variable on either side
of the equal sign.
Example
3: Check
$\eqalign{3y + 8 – 5y =& 4
&{\small\color{red}\quad\quad\text{Write original
equation.}} \cr 3\left( \color{red}2\color{black} \right) + 8 –
5\left(\color{red}2\color{black} \right)
\overset{?}{=}& 4
&{\small\color{red}\quad\quad\text{Substitute 2
for }x.}
\cr 6 + 8 – 10\overset{?}{=}& 4
&{\small\color{red}\quad\quad\text{Multiply.}} \cr4 =&
4&{\small\color{red}\quad\quad\text{Solution
checks. }\checkmark} \cr} $
Study
Tip
When you are solving an equation, you can interpret your results as
follows.
| Result | Example | Conclustion |
| An equation that is true for one value |
$x = 4$ $-3 = x$ |
The equation has exactly one solution. |
| An equation that is always true |
$1 = 1$ $x = x$ |
The equation has infinitely many solutions. |
| An equation that is always false |
$7 = 0$ $-2 = 2$ |
The equation has no solution. |
Example
4: Check
a.
There is no solution to check. There
is no value of x
that will make 3 equal to 8.
b.
The equation is true for any value of
x. You can
test this by picking a few values of x and checking if
they
satisfy the equation. This won’t prove that the solution is correct,
but it will give an indication of the validity of the solution. For
instance, try $x=0$ and $x=5$.
Let $x = 0:$
$\eqalign{4\left( {x + 3} \right)
=& 4x + 12&{\small\color{red}\quad\quad\text{Write
original equation.}} \cr4\left( { \color{red}0\color{black}
+ 3} \right)\overset{?}{=}&
4\left(\color{red}0\color{black}
\right) + 12&{\small\color{red}\quad\quad\text{Substitute 0
for }x.} \cr 4\left( 3
\right)\overset{?}{=}& 0 +
12&{\small\color{red}\quad\quad\text{Simplify.}}
\cr 12 =& 12&{\small\color{red}\quad\quad\text{Solution
checks. }\checkmark} \cr}$
Let $x = 5:$
$\eqalign{4\left( {x + 3} \right)
=& 4x + 12&{\small\color{red}\quad\quad\text{Write
original equation.}} \cr4\left( { \color{red}5\color{black}
+ 3} \right)\overset{?}{=}&
4\left(\color{red}5\color{black}
\right) + 12&{\small\color{red}\quad\quad\text{Substitute 5
for }x.} \cr 4\left( 8
\right)\overset{?}{=}& 20 +
12&{\small\color{red}\quad\quad\text{Simplify.}}
\cr 32 =& 32&{\small\color{red}\quad\quad\text{Solution
checks. }\checkmark} \cr}$