Equations in Algebra Fractions

This is a type of Algebra fractions that would lead into a linear or quadratic equation after applying some basic operations like cross-multiplication, L.C.M, etc...

Note; Whenever you have a fraction equated to a value, whether single digit, two digit or more, it is advisable to cross-multiply by putting the single or two digit value over 1;

Examples; Solve the following;

Solution

Step 1

Note; The question above has a fraction at the LHS

and a two digit value a – 2 at the RHS.

Make the two digits value a fraction by putting it over 1 and hence cross-multiply both fractions together. See below!

Step 2

Cross – Multiply;

a(a – 2) = 3 × 1

a(a – 2) = 3

Step 3

Remove bracket;

a² – 2a = 3

a² – 2a – 3 = 0  ---this has turned into a quadratic equation.

Step 4

Factorise the equation;

a² – 2a – 3 = 0             -3a² (×)      (+3a & -1a)

                                        -2a (+)

a² + 3a – 1a – 3 = 0

a(a + 3) -1(a + 3) = 0

(a + 3)(a – 1) = 0

a + 3 = 0     OR     a – 1 = 0

a = 0 – 3     OR     a = 0 + 1

a = 3     OR     a = 1

Step 1

Put 5 – 2d over 1;

Step 2

Cross – Multiply;

2 × 1 = d(5 – 2d)

Step 3

Remove bracket;

2 = 5d – 2d²

Step 4

Rearrange;

2d² – 5d + 2 = 0  ---this is a quadratic equation.

Step 5

Factorise the equation;

2d² – 5d + 2 = 0            +4d² (×)     (-4d & -1d)

                                          -5d (+)

2d² -4d – 1d + 2 = 0

2d(d – 2) – 1(d – 2) = 0

(d – 2)(2d – 1) = 0

d – 2 = 0     OR     2d – 1 = 0

d = 0 + 2     OR     2d = 0 + 1

d = 2     OR     2d = 1

d = 2     OR   2d/2 = 1/2

d = 2     OR     d = 1/2

Step 1

Note; This particular question has two added fraction at the LHS, so we will first find the L.C.M of these fractions before we can cross multiply the sides;

The L.C.M will be “e”,

Step 2

Put “e” over 1;

Step 3

Cross – Multiply;

e × e = 7 + 2

e² = 9  ---this is a simple (linear) equation.

Step 4

Take square root of both sides;

e = 3

Step 1

Put c over 1;

Step 2

Cross – Multiply;

3 × 1 = c(c + 2)

Step 3

Remove bracket;

3 = c² + 2c

Step 4

Rearrange;

-c² – 2c + 3 = 0  ---this is a quadratic equation.

Step 5

Factorise the equation;

-c² – 2c + 3 = 0          -3c² (×)      (-3c & +1c)

                                      -2c (+)

-c² – 3c + 1c + 3 = 0

-c(c + 3) + 1(c + 3) = 0

(c + 3)(-c + 1) = 0

c + 3 = 0     OR     -c + 1 = 0

c = 0 – 3     OR     -c = 0 – 1

c = 3     OR     -c = -1 (multiply both sides by -1)

c = 3     OR     (-c × -1) = (-1 × -1)

c = 3     OR     c = 1

Step 1

Put m over 1;

Step 2

Cross – Multiply;

8 × 1 = m(3m + 2)

Step 3

Remove bracket;

8 = 3m² + 2m

Step 4

Rearrange;

-3m² – 2m + 8 = 0  ---this is a quadratic equation.

Step 5

Factorise the equation;

-3m² – 2m + 8 = 0        -24m² (×)      (-6m & +4m)

                                          -2m (+)

-3m² – 6m + 4m + 8 = 0

-3m(m + 2) + 4(m + 2) = 0

(m + 2)(-3m + 4) = 0

m + 2 = 0     OR     -3m + 4 = 0

m = 0 – 2     OR     -3m = 0 – 4

m = -2     OR     -3m = -4 (divide both sides by -3)

m = -2     OR    -3m/-3 = -4/-3

m = -2     OR     m = 4/3 (convert to mixed fraction)

m = -2     OR     m = 1^1/3