Problem 10 - Algebra 1

Question

Solve the equation. $$-\frac{3}{8} t=-6$$

Step-by-Step Solution

Answer

The solution to the equation is t = 16.

Step 1: Identify the equation

The equation given is -\(\frac{3}{8}\)t = -6 which we need to solve for 't'.

Step 2: Isolating the term containing 't'

To isolate 't', we need to get rid of the coefficient -\(\frac{3}{8}\). We do this by multiplying both sides of the equation by its reciprocal, in this case -\(\frac{8}{3}\).

Step 3: Solve for 't'

Multiplying both sides of the equation by -\(\frac{8}{3}\) gives: t = 16.

Key Concepts

Solving Equations

When you solve an equation, you are essentially finding a value for the variable that makes the equation true. In the given problem, we have a linear equation, which is an equation of the first degree.

• The main goal here is to find the value of \(t\) such that the left and right sides of the equation are equal.
• This often involves performing the same operation on both sides to maintain the balance of the equation.

Linear equations are straightforward because they typically involve only addition, subtraction, multiplication, or division of variables and numbers. In order to solve them:

• You start by understanding the operations that are being performed on the variable.
• The next step is to reverse these operations to isolate the variable and solve for it.
• This often involves adding or subtracting terms from both sides, or multiplying or dividing both sides by a number.

Remember, whatever you do to one side of the equation, you must do to the other to keep the equation balanced.

Reciprocal

The concept of a reciprocal is vital when dealing with equations that involve fractions. The reciprocal of a fraction is simply flipping the numerator and the denominator. In this problem, the coefficient of \(t\) was \(-\frac{3}{8}\), so its reciprocal would be \(-\frac{8}{3}\).

• Using the reciprocal helps in clearing fractions in an equation, especially when you need to isolate variables.
• When you multiply a fraction by its reciprocal, you are left with 1, which effectively removes the fraction.

For instance, if the fraction is negative as in our case, the reciprocal will also be negative to ensure balance when multiplied. Applying the reciprocal:

• Turn \(\frac{a}{b}\) into \(\frac{b}{a}\).
• Use it to simplify the equation, making calculations much simpler and straightforward.

This step significantly aids in isolating the variable quickly and effectively in equations.

Isolation of Variable

To solve for a variable, you first need to isolate it. This typically involves moving all other numbers and expressions to the opposite side of the equation. In this exercise, isolating \(t\) was possible by using the reciprocal of its coefficient.

• The idea is to eventually have the variable on one side and the numbers on the other.
• By multiplying both sides of the equation by \(-\frac{8}{3}\), the initially complex fractional coefficient of \(t\) disappeared, leaving \(t\) on its own.

What you did was eliminate the term attached to \(t\), making it possible to directly solve for it. Remember:

• Perform operations that simplify the equation and clearly show the value of the variable.
• Work step-by-step, focusing on eliminating coefficients, fractions, or any terms attached to the variable until it's isolated.

Once \(t\) is isolated, you just need to perform the straightforward arithmetic on the remaining numbers to find its value. This method is essential in making complex equations more manageable.