Solve \frac{t}{t-3}-\frac{5}{4t-12}

Question

Simplify the expression

\frac{4 t - 5}{4 t - 12}

Evaluate

\frac{t}{t - 3} - \frac{5}{4 t - 12}

Factor the expression

\frac{t}{t - 3} - \frac{5}{4 ( t - 3 )}

Reduce fractions to a common denominator

\frac{t \times 4}{( t - 3 ) \times 4} - \frac{5}{4 ( t - 3 )}

Use the commutative property to reorder the terms

\frac{t \times 4}{4 ( t - 3 )} - \frac{5}{4 ( t - 3 )}

Write all numerators above the common denominator

\frac{t \times 4 - 5}{4 ( t - 3 )}

Use the commutative property to reorder the terms

\frac{4 t - 5}{4 ( t - 3 )}

Solution

More Steps

Evaluate

4 (t - 3)

Apply the distributive property

4 t - 4 \times 3

Multiply the numbers

4 t - 12

\frac{4 t - 5}{4 t - 12}

Show Solution

Find the excluded values

t = 3

Evaluate

\frac{t}{t - 3} - \frac{5}{4 t - 12}

To find the excluded values,set the denominators equal to 0

t - 3 = 0 4 t - 12 = 0

Solve the equations

More Steps

Evaluate

t - 3 = 0

Move the constant to the right-hand side and change its sign

t = 0 + 3

Removing 0 doesn't change the value,so remove it from the expression

t = 3

t = 3 4 t - 12 = 0

Solve the equations

More Steps

Evaluate

4 t - 12 = 0

Move the constant to the right-hand side and change its sign

4 t = 0 + 12

Removing 0 doesn't change the value,so remove it from the expression

4 t = 12

Divide both sides

\frac{4 t}{4} = \frac{12}{4}

Divide the numbers

t = \frac{12}{4}

Divide the numbers

More Steps

Evaluate

\frac{12}{4}

Reduce the numbers

\frac{3}{1}

Calculate

3

t = 3

t = 3 t = 3

Solution

t = 3

Show Solution

Find the roots

t = \frac{5}{4}

Alternative Form

t = 1.25

Evaluate

\frac{t}{t - 3} - \frac{5}{4 t - 12}

To find the roots of the expression,set the expression equal to 0

\frac{t}{t - 3} - \frac{5}{4 t - 12} = 0

Find the domain

More Steps

Evaluate

{t - 3 \neq = 0 4 t - 12 \neq = 0

Calculate

More Steps

Evaluate

t - 3 \neq = 0

Move the constant to the right side

t \neq = 0 + 3

Removing 0 doesn't change the value,so remove it from the expression

t \neq = 3

{t \neq = 3 4 t - 12 \neq = 0

Calculate

More Steps

Evaluate

4 t - 12 \neq = 0

Move the constant to the right side

4 t \neq = 0 + 12

Removing 0 doesn't change the value,so remove it from the expression

4 t \neq = 12

Divide both sides

\frac{4 t}{4} \neq = \frac{12}{4}

Divide the numbers

t \neq = \frac{12}{4}

Divide the numbers

t \neq = 3

{t \neq = 3 t \neq = 3

Find the intersection

t \neq = 3

\frac{t}{t - 3} - \frac{5}{4 t - 12} = 0, t \neq = 3

Calculate

\frac{t}{t - 3} - \frac{5}{4 t - 12} = 0

Subtract the terms

More Steps

Simplify

\frac{t}{t - 3} - \frac{5}{4 t - 12}

Factor the expression

\frac{t}{t - 3} - \frac{5}{4 ( t - 3 )}

Reduce fractions to a common denominator

\frac{t \times 4}{( t - 3 ) \times 4} - \frac{5}{4 ( t - 3 )}

Use the commutative property to reorder the terms

\frac{t \times 4}{4 ( t - 3 )} - \frac{5}{4 ( t - 3 )}

Write all numerators above the common denominator

\frac{t \times 4 - 5}{4 ( t - 3 )}

Use the commutative property to reorder the terms

\frac{4 t - 5}{4 ( t - 3 )}

\frac{4 t - 5}{4 ( t - 3 )} = 0

Cross multiply

4 t - 5 = 4 (t - 3) \times 0

Simplify the equation

4 t - 5 = 0

Move the constant to the right side

4 t = 0 + 5

Removing 0 doesn't change the value,so remove it from the expression

4 t = 5

Divide both sides

\frac{4 t}{4} = \frac{5}{4}

Divide the numbers

t = \frac{5}{4}

Check if the solution is in the defined range

t = \frac{5}{4}, t \neq = 3

Solution

t = \frac{5}{4}

Alternative Form

t = 1.25

Show Solution