Question

Solve and check each equation. Treat the constants in these equations as exact numbers. Leave your answers in fractional, rather than decimal, form. $$4 t+9=11 t-3 t$$

Short answer

\(t = \frac{9}{4}\)

Step-by-step solution

Combine Like Terms on One Side

Subtract the term with the variable from both sides to collect like terms on one side. This results in: \(4t - (11t - 3t) = -9\).

Simplify Both Sides

Simplify the equation by combining like terms. Subtract \(11t\) and \(3t\) to get: \(4t - 11t + 3t = -9\), which simplifies to \(-4t = -9\).

Divide by the Coefficient of the Variable

Divide both sides of the equation by \(-4\) to isolate \(t\): \(t = \frac{-9}{-4}\).

Simplify the Fraction

The negative signs cancel out, leaving the solution as: \(t = \frac{9}{4}\).

Check the Solution

Replace \(t\) in the original equation with \(\frac{9}{4}\) to check if the equation is true: \(4\left(\frac{9}{4}\right) + 9 = 11\left(\frac{9}{4}\right) - 3\left(\frac{9}{4}\right)\). Simplifying, we get \(9 + 9 = 9 \times 2\), which is a true statement.

Key concepts

Combining Like Terms

When solving linear equations, one of the first steps often involves combining like terms. Like terms are terms that have the same variable raised to the same power. The reason we combine them is to simplify the equation and make it more manageable.

For example, in the equation from the exercise, we start with terms that include the variable 't': \(4t+9=11t-3t\). The '11t' and '-3t' are like terms, and combining them simplifies our equation. To combine them, you subtract '3t' from '11t', giving you '8t'. This is how you reduce the equation to fewer terms, making the next steps of solving it more straightforward.

This process is similar to consolidating items when packing; grouping similar items together saves space and makes it easier to account for everything you have, just as combining like terms simplifies equations.

Isolation of Variables

Isolation of the variable is a crucial step in solving linear equations. It means rearranging the equation so that the variable we want to solve for, say 't', stands alone on one side of the equation. This step is essential to determine the value of the variable.

Referencing our exercise, after combining like terms, we've reached the interim step of \(4t - 11t + 3t = -9\), which simplifies to \( -4t = -9\). The next move is to isolate 't.' We do this by dividing both sides of the equation by '-4', the coefficient of 't.' This gives us \(t = \frac{-9}{-4}\), achieving the isolation of 't.'

It's similar to clearing a space on a crowded desk so that only one item, the one we're focusing on, is in the center. Everything else is moved aside.

Fractional Solutions

Fractional solutions are common in linear equations, especially when the coefficients of variables and constants do not divide evenly. These solutions are perfectly valid and can provide exact values.

In our exercise, once we isolated the variable 't', we found that the solution took the form of a fraction: \(t = \frac{-9}{-4}\). After simplifying the fraction by dividing both the numerator and the denominator by their greatest common divisor or by canceling out common factors, we have our solution: \(t = \frac{9}{4}\).

Just like in recipe measurements, fractional solutions give us the precise amount needed—no more, no less. They are the exact 'ingredients' for our equation.

Equation Checking

Equation checking is the final step in solving linear equations. This step is crucial because it helps to verify that the obtained solution is indeed correct. After solving for the variable, we substitute it back into the original equation to see if it balances.

In this exercise, after finding that \(t = \frac{9}{4}\), we substitute this value back into the original equation: \(4\left(\frac{9}{4}\right) + 9 = 11\left(\frac{9}{4}\right) - 3\left(\frac{9}{4}\right)\). Simplifying both sides ensures that they are equal, which confirms that \(\frac{9}{4}\) is indeed the correct solution.

It's much like re-reading a written set of directions to ensure you've followed the steps correctly and have arrived at the right destination. Checking your equations reaffirms your journey from problem to solution.