Question

Solve each equation, if possible. $$ \frac{6 t+7}{4 t-1}=\frac{3 t+8}{2 t-4} $$

Short answer

\(t = \frac{-20}{39}\)

Step-by-step solution

Cross Multiply

To solve the equation \( \frac{6t + 7}{4t - 1} = \frac{3t + 8}{2t - 4} \), we will start by cross-multiplying to eliminate the fractions. This means multiplying both sides of the equation by \((4t - 1)(2t - 4)\). Cross-Multiplication gives us: \((6t + 7)(2t - 4) = (3t + 8)(4t - 1)\).

Distribute and Expand

Now, distribute the terms in each product. For the left-hand side: \((6t + 7)(2t - 4)\), we get:\[ 6t \times 2t + 6t \times -4 + 7 \times 2t + 7 \times -4 \] which simplifies to:\[ 12t^2 - 24t + 14t - 28 \] Combining like terms yields:\[ 12t^2 - 10t - 28 \]. For the right-hand side: \((3t + 8)(4t - 1)\), we get:\[ 3t \times 4t + 3t \times -1 + 8 \times 4t + 8 \times -1 \] which simplifies to:\[ 12t^2 - 3t + 32t - 8 \] Combining like terms yields:\[ 12t^2 + 29t - 8 \].

Set the Equation Equal and Simplify

Now set the expanded equations equal to one another and simplify:\[ 12t^2 - 10t - 28 = 12t^2 + 29t - 8 \].Subtract \(12t^2\) from both sides to eliminate the quadratic terms:\[ -10t - 28 = 29t - 8 \].

Solve for t

To isolate \(t\), first add \(10t\) to both sides:\[ -28 = 39t - 8 \]Next, add \(8\) to both sides:\[ -20 = 39t \]Finally, divide both sides by 39:\[ t = \frac{-20}{39} \].

Verify the Solution

To ensure the solution is correct, substitute \( t = \frac{-20}{39} \) back into the original equation and check both sides. If both sides are equal, the solution is verified.

Key concepts

Cross Multiplication

Rational equations often involve fractions, making it challenging to solve them directly. Cross-multiplication is a crucial technique. When faced with an equation like \( \frac{6t + 7}{4t - 1} = \frac{3t + 8}{2t - 4} \), it's helpful to eliminate the fractions.
To do this, you multiply the numerator of each fraction by the denominator of the other fraction. This gives: \( (6t + 7)(2t - 4) = (3t + 8)(4t - 1) \).
This step simplifies the equation by getting rid of the fractions, making it easier to handle.

Distributing and Combining Like Terms

Once you perform cross multiplication, the next step is distributing the terms. Let's take the left-hand side first:
\[ (6t + 7)(2t - 4) \] Distributing gives:
\[ 6t \times 2t + 6t \times -4 + 7 \times 2t + 7 \times -4 \ = 12t^2 - 24t + 14t - 28 \]
Combining like terms results in:
\[ 12t^2 - 10t - 28 \]
Now, for the right-hand side: \((3t + 8)(4t - 1)\)
Distributing gives:
\[ 3t \times 4t + 3t \times -1 + 8 \times 4t + 8 \times -1 \ = 12t^2 - 3t + 32t - 8 \]
Combining like terms results in:
\[ 12t^2 + 29t - 8 \]
Distributing and combining terms not only simplifies the equation but also makes the variables clear and manageable.

Isolating Variables

After simplifying both sides, you need to set the equations equal and isolate the variable. For example, you get:
\[ 12t^2 - 10t - 28 = 12t^2 + 29t - 8 \]
Subtract \(12t^2\) from both sides:
\[ -10t - 28 = 29t - 8 \]
Next, add \(10t\) to both sides to further isolate \(t\):
\[ -28 = 39t - 8 \]
Finally, add \(8\) to both sides:
\[ -20 = 39t \]
To solve for \(t\), divide both sides by \(39\):
\[ t = \frac{-20}{39} \]
Isolating the variable step-by-step shrinks the equation, making it easier to solve.

Verifying Solutions

To verify your solution, always substitute it back into the original equation. Let's test \( t = \frac{-20}{39} \). Replace \(t\) in the original equation:
\[ \frac{6(\frac{-20}{39}) + 7}{4(\frac{-20}{39}) - 1} = \frac{3(\frac{-20}{39}) + 8}{2(\frac{-20}{39}) - 4} \]
Simplify both sides to ensure they are equal.
This final step is crucial. It confirms the accuracy of the solution, ensuring you haven't made errors in your calculations.