Linear Equations with Fractions

Undo the division by multiplying both sides by the denominator.

1. Multiply both sides by 5.
2. This gives x=15.

Clear the denominator before subtracting the constant.

1. Multiply both sides by 3 to get x+4=21.
2. Subtract 4 to get x=17.

Multiply both sides by the denominator, then collect the constant terms.

1. Multiply both sides by 5 to get 2x-1=15.
2. Add 1 to get 2x=16, so x=8.

Isolate the fractional term first.

1. Subtract 2 from both sides to get \dfrac x4=3.
2. Multiply by 4 to get x=12.

After clearing the denominator, keep the negative sign attached to the unknown.

1. Multiply by 2 to get 7-x=12.
2. Subtract 7 to get -x=5, so x=-5.

Multiply every term by a common multiple of both denominators.

1. The lowest common multiple of 3 and 5 is 15, so multiply every term by 15.
2. 15\times\dfrac x3=5x and 15\times\dfrac x5=3x, so the equation becomes 5x+3x=120.
3. So 8x=120 and x=15.

Multiply both sides by a common denominator and keep each numerator in brackets.

1. Multiply by 20 to get 5(x+1)=4(2x-3).
2. Expand to get 5x+5=8x-12.
3. Collect terms to get 17=3x, so x=\dfrac{17}{3}.

Multiply the whole equation by four, including the integer term.

1. Multiply every term by 4: 3x+2-4=2(x-1).
2. Simplify to get 3x-2=2x-2.
3. Subtract 2x and add 2 to get x=0.

The minus sign applies to the whole second numerator.

1. The lowest common multiple of 6 and 3 is 6. Multiply every term by 6, keeping the second numerator in a bracket: 5x-2-2(x+4)=6.
2. Expand the bracket. The minus sign changes both signs inside it: 5x-2-2x-8=6.
3. Collect terms: 3x-10=6, so 3x=16.
4. So x=\dfrac{16}{3}.

Clear all three numerical denominators at once.

1. Multiply every term by 12 to get 9x-8=10.
2. Add 8 to get 9x=18.
3. Divide by 9 to get x=2.

Write the difference between the two fractional amounts as an equation.

1. Let the number be n. Then \dfrac n3-\dfrac n5=6.
2. Multiply by 15 to get 5n-3n=90.
3. So 2n=90 and the number is 45.

Express the amount remaining in terms of the original funds.

1. Let the original funds be x pounds. After a quarter is spent on posters, \dfrac34x is left.
2. After the £18 for printing, \dfrac34x-18 is left. This equals two thirds of the original funds: \dfrac34x-18=\dfrac23x.
3. Multiply every term by 12, the lowest common multiple of 4 and 3: 9x-216=8x.
4. Subtract 8x and add 216 to both sides: x=216. The charity had £216 originally.

Write the larger even integer in terms of the smaller one.

1. Let the smaller even integer be n, so the larger is n+2.
2. Then \dfrac{n+2}{2}+\dfrac n3=21.
3. Multiply by 6: 3(n+2)+2n=126.
4. So 5n+6=126, giving the smaller integer n=24.

Translate the relationship into an equation before clearing its denominators.

1. A is twice B means A=2B: \dfrac{3x+7}{4}=\dfrac{2(x-1)}{3}.
2. Multiply both sides by 12: 3(3x+7)=4\times2(x-1), which is 3(3x+7)=8(x-1).
3. Expand: 9x+21=8x-8.
4. Subtract 8x and 21 from both sides: x=-29.

Replace both function terms with the expressions given by the rule.

1. Replace x by x+3 in the rule: f(x+3)=\dfrac{2(x+3)-5}{3}=\dfrac{2x+1}{3}.
2. The equation becomes \dfrac{2x-5}{3}+\dfrac{2x+1}{3}=7.
3. Multiply every term by 3: 2x-5+2x+1=21, so 4x-4=21.
4. So 4x=25 and x=\dfrac{25}{4}.

Use a common denominator that clears all three fractions.

1. Multiply every term by 20: 4(2x+3)+10(x-4)=5(3x+1).
2. Expand to get 8x+12+10x-40=15x+5.
3. Collect terms: 18x-28=15x+5.
4. So 3x=33 and x=11.

Multiply through by the common denominator before expanding the brackets.

1. Multiply by 12: 8(5-x)-3(x+2)=10.
2. Expand carefully: 40-8x-3x-6=10.
3. This simplifies to 34-11x=10.
4. Therefore 11x=24 and x=\dfrac{24}{11}.

Represent both colour totals using one unknown before writing an equation for the sample.

1. Let the number of blue counters be b. The number of red counters is b+12.
2. The sample gives \dfrac{b+12}{3}+\dfrac b4=18.
3. Multiply by 12: 4(b+12)+3b=216.
4. Then 7b+48=216, so b=24 and there are 36 red counters.
5. The total number of counters is 24+36=60.

Get the fraction containing x on its own first, then multiply both sides by its denominator.

1. Add 2 to both sides. Since 2+\dfrac14=\dfrac94, this gives \dfrac{3x-1}{x+2}=\dfrac94.
2. Multiply both sides by 4(x+2) to clear both denominators: 4(3x-1)=9(x+2).
3. Expand: 12x-4=9x+18.
4. Subtract 9x and add 4 to both sides: 3x=22, so x=\dfrac{22}{3}.

Multiply every term by the lowest common denominator, including the constant fraction.

1. The lowest common multiple of 8, 6, 3 and 4 is 24. Multiply every term by 24, keeping each numerator in a bracket: 3(5x+1)-4(x+2)=8(x-2)+6.
2. Expand the brackets: 15x+3-4x-8=8x-16+6.
3. Collect terms on each side: 11x-5=8x-10.
4. Subtract 8x and add 5 to both sides: 3x=-5, so x=-\dfrac53.