Solving Linear Equations with Fractions

Linear equations have variables without exponents. Fractions occur in linear equations as coefficients of the variables or as constants.Students are generally able to solve linear equations without fractions with ease using properties of equality. But linear equations with fractions appear more complicated and pose some difficulty in solving them
While solving linear equations we eliminate fractions first whenever it is possible or required to do so. But fractions in some specific form of linear equations have a definite significance.
Fractions are integral part of math and acquiring the skill to handle them is a must to attain a high degree of proficiency in Math.

Eliminating a fraction by multiplication

The fractions in an equation are eliminated by multiplying the equation with the LCM of the denominators present in the fractions.This is a general method and can be used to eliminate fractions in any equation.

Example:
Solve for x:  $\frac{1}{4}x+\frac{1}{6}=\frac{1}{9}x+6$
The equation contains 3 fractions $\frac{1}{4}$, $\frac{1}{6}$ and $\frac{1}{9}$.  The denominators of the fractions are 4,6 and 9.
The LCM of 4,6 and 9 = 36.  So the equation has to be multiplied by 36 to get rid of the denominators.
$9x+6 = 4x + 216$                                           Distribution of 36 and elimination of fractions.
$-6 -6$                                            Subtraction Property
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$9x = 4x + 210$
$-4x -4x$                                                    Subtraction Property
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$5x = 210$
$\frac{5x}{5}=\frac{210}{5}$                          Division Property
$x= 42$                                                  Solution of the equation

Linear equations with two variables.

Fractions in some forms of linear equations are retained as they indicate some parameter values.
For example, in the slope intercept form of the following equation,
$y =\frac{2}{3}x -\frac{1}{2}$, the fractions are not eliminated if we want to know the slope or y intercept of the equation. Comparing with the general equation for slope intercept for $y=mx+b$, the slope of the line is given by m = $\frac{2}{3}$ and the y intercept by b = $-\frac{1}{2}$.

Similarly the intercepts form of a linear equation $\frac{x}{a}+\frac{y}{b}=1$ tells that $a$ and $b$ are the x and y intercepts of the line.
So if we have the equation of a line as $\frac{x}{4}+\frac{y}{-5}=1$, we can conclude that the x and y intercepts of the line are 4 and -5 respectively.