In this chapter we will learn cross multiplication method to solve linear equation with solved examples.
What is cross multiplication method ?
When two algebraic fractions are present on either side of the equation, then the equation can be solved by;
⟹ multiplying denominator on left to numerator on right.
⟹ multiplying denominator on right with numerator on left.
The general expression of cross multiplication is given as;
\mathtt{\Longrightarrow \ \frac{a}{b} \ =\ \frac{c}{d}}\\\ \\ \mathtt{\Longrightarrow \ a\ \times \ d\ =\ c\ \times \ b}
I hope you understood the process. Let us solve some examples for further clarity.
Solving equations using cross multiplication
Example 01
Solve the below equation and find value of x.
\mathtt{\Longrightarrow \ \frac{x}{2} \ =\ \frac{7}{8}}
Solution
Cross multiplying the expressions we get;
\mathtt{\Longrightarrow \ 8\times x\ =\ 7\ \times 2}\\\ \\ \mathtt{\Longrightarrow \ 8x\ =\ 14}\\\ \\ \mathtt{\Longrightarrow \ x\ =\frac{14}{8}}
The value of x can be further simplified by dividing numerator and denominator by 2.
\mathtt{\Longrightarrow \ x\ =\frac{14\div 2}{8\div 2}}\\\ \\ \mathtt{\Longrightarrow \ x\ =\ \frac{7}{4}}
Hence, 7/4 is the value of x.
Example 02
Solve the equation and find value of x
\mathtt{\Longrightarrow \ \frac{x+3}{5} \ =\ \frac{x+2}{4}}
Solution
Cross multiplying the numbers we get;
\mathtt{\Longrightarrow \ 4\ ( x\ +3) \ =\ 5\ ( x\ +2)}\\\ \\ \mathtt{\Longrightarrow \ 4x\ +\ 12\ =\ 5x\ +\ 10}\\\ \\ \mathtt{\Longrightarrow \ 12-\ 10\ =\ 5x\ -\ 4x}\\\ \\ \mathtt{\Longrightarrow \ 2=\ x}\\\ \\ \mathtt{\Longrightarrow \ x\ =\ 2\ }
Hence, x = 2 is the solution of given expression.
Example 03
Solve the below equation.
\mathtt{\Longrightarrow \ \frac{x}{2} +\frac{x}{5} \ =\ \frac{4}{7}}
Solution
Here the solution involves addition of fraction.
To learn how to add two fractions, click the red link.
\mathtt{\Longrightarrow \ \frac{5x+2x}{10} =\ \frac{4}{7}}\\\ \\ \mathtt{\Longrightarrow \ \frac{7x}{10} =\ \frac{4}{7}}
Now cross multiplying numbers, we get;
\mathtt{\Longrightarrow \ \frac{7x}{10} =\ \frac{4}{7}}\\\ \\ \mathtt{\Longrightarrow \ 7x\times 7=\ 4\ \times 10}\\\ \\ \mathtt{\Longrightarrow \ 49x\ =\ 40}\\\ \\ \mathtt{\Longrightarrow \ x\ =\frac{40}{49}}
Hence, value of x is 40/49
Example 04
Solve linear equation and find value of x.
\mathtt{\Longrightarrow \ \frac{11x-4}{2x+13} =\ 2}
Solution
The above expression can be written as;
\mathtt{\Longrightarrow \ \frac{11x-4}{2x+13} =\ \frac{2}{1}}
Now cross multiplying the numbers we get;
\mathtt{\Longrightarrow \ ( 11x-4) \times 1=\ 2\times ( 2x+13)}\\\ \\ \mathtt{\Longrightarrow \ 11x-4\ =\ 4x\ +\ 26}\\\ \\ \mathtt{\Longrightarrow \ 11x-4x=26+4}\\\ \\ \mathtt{\Longrightarrow \ 7x=30}\\\ \\ \mathtt{\Longrightarrow \ x=\frac{30}{7}}
Hence, value of x is 30 / 7.
Example 05
Solve the equation and find value of x.
\mathtt{\Longrightarrow \ \frac{x}{2} +\frac{x}{3} +4=2x-3\ }
Solution
\mathtt{\Longrightarrow \ \frac{x}{2} +\frac{x}{3} =2x-3-4}\\\ \\ \mathtt{\Longrightarrow \ \frac{3x+2x}{6} =2x-7}\\\ \\ \mathtt{\Longrightarrow \ \frac{3x+2x}{6} =\frac{2x-7}{1}}
Cross multiplying the numbers we get;
\mathtt{\Longrightarrow \ ( 3x+2x) \times 1\ =\ ( 2x-7) \times 6}\\\ \\ \mathtt{\Longrightarrow \ 3x+2x=12x-42}\\\ \\ \mathtt{\Longrightarrow \ 5x\ =\ 12x-42}\\\ \\ \mathtt{\Longrightarrow 5x-12x\ =\ -42}\\\ \\ \mathtt{\Longrightarrow \ -7x\ =\ -42}\\\ \\ \mathtt{\Longrightarrow \ x\ =\frac{-42}{-7}}\\\ \\ \mathtt{\Longrightarrow \ x\ =\ 6}
Hence, x = 6 is the solution of given problem.