Problem 45 - Algebra 1

Question

Solve the equation. $$\frac{3}{4} t=|-15|$$

Step-by-Step Solution

Answer

The solution to the equation is \(t = 20\).

Step 1: Understand the Absolute Value

First, understand what the absolute value means. It is the distance of a number from zero, regardless of direction. So, the absolute value of any number is always positive or zero. Therefore, the absolute value of -15 is 15.

Step 2: Substituting the Absolute Value

Substitute the absolute value of -15 with 15 in the equation. Hence, the equation becomes \(\frac{3}{4} t=15\).

Step 3: Solve for t

Finally, solve for t by dividing both sides of the equation by \(\frac{3}{4}\). This can be rewritten as multiplying the both sides by the reciprocal of \(\frac{3}{4}\), which is \(\frac{4}{3}\). Doing so gives the final solution: \(t = 15 \times \frac{4}{3} = 20\).

Key Concepts

Absolute Value

When we refer to the 'absolute value' of a number, we're talking about the number's distance from zero on the number line, without considering which direction it's in. Let's illustrate this with a simple example. The absolute value of -3 is 3, because if you start at zero and move 3 units to the left (the direction for negative numbers), you've traveled a distance of 3 units. The same is true if you move 3 units to the right (the direction for positive numbers).

So, in the context of our exercise, when you see an equation that includes absolute value, such as \( \frac{3}{4} t = |-15| \), it implies that the value within the absolute value sign should be considered as a positive distance. Regardless of the fact that -15 is negative, the absolute value of -15 is 15, just like the absolute value of +15 is 15 because both are 15 units away from zero on the number line.

Algebraic Expressions

Algebraic expressions are like puzzles in the world of mathematics, where letters often represent numbers, and we have to figure out the value of these letters. In algebra, these letters, known as variables, can be manipulated using arithmetic operations to solve for their values.

For instance, consider the expression \( \frac{3}{4} t \). Here, \(t\) is a variable, and it’s being multiplied by \( \frac{3}{4} \). To find the value of \(t\), we need to solve the equation by performing algebraic operations that will isolate \(t\) on one side of the equation. This concept is crucial to understanding how to move from an algebraic expression to a solution.

Equation Solving

Solving equations is a fundamental skill in algebra. The goal is to isolate the variable in question on one side of the equation, enabling us to find its value. This process usually involves performing the same mathematical operations on both sides of the equation to maintain equality.

For our exercise, we need to solve for \(t\) in the equation \( \frac{3}{4} t = 15 \). To do this, we must perform operations that will leave \(t\) by itself. This often involves 'undoing' whatever operation is being done to the variable. Since \(t\) is being multiplied by \( \frac{3}{4} \), we can undo this multiplication by using its opposite operation, which is division. Alternatively, we can multiply by the reciprocal, which is also a fundamental process in algebra known as 'reciprocal multiplication'.

Reciprocal Multiplication

Reciprocal multiplication involves multiplying a number by the reciprocal (or inverse) of another number. The reciprocal of a fraction \( \frac{a}{b} \) is \( \frac{b}{a} \), and for any non-zero number \(x\), its reciprocal is \( \frac{1}{x}\). When a number is multiplied by its reciprocal, the result is always 1, which is why we use this technique to cancel out coefficients in equations.

In our exercise, to isolate \(t\), we multiply both sides by the reciprocal of \( \frac{3}{4} \) which is \( \frac{4}{3} \). This action cancels out the \( \frac{3}{4}\) on the left side and leaves us with the variable \(t\) on its own, allowing us to find its value. It's a neat trick that helps us move from an equation to its solution efficiently.