Problem 60 - Prealgebra

Question

Solve each equation. $$7 t=42$$

Step-by-Step Solution

Answer

The solution for \(t\) is 6.

Step 1: Understand the Equation

We are given the equation \(7t = 42\). This is a simple linear equation where \(t\) is being multiplied by 7.

Step 2: Isolate the Variable

To solve for \(t\), we need to isolate it on one side of the equation. Since \(t\) is multiplied by 7, we can isolate \(t\) by dividing both sides of the equation by 7.

Step 3: Perform the Division

Divide both sides of the equation by 7: \(\frac{7t}{7} = \frac{42}{7}\). This simplifies the left side to \(t\) and the right side to 6.

Step 4: Simplify the Equation

After the division, we have \(t = 6\). This means that when \(t\) is 6, the equation \(7t = 42\) holds true.

Key Concepts

Isolating the Variable

When solving equations, one of the main goals is to isolate the variable. This means getting the variable on one side of the equation all by itself. In the equation given, \( 7t = 42 \), the variable \( t \) is currently being multiplied by 7. If we want to find the value of \( t \), we must "undo" what's being done to it first.
To isolate \( t \):

• We need to do the opposite of the operation currently affecting \( t \). Here, it is being multiplied by 7, so we will use division to "cancel out" the 7.
• In doing so, remember whatever operation you apply to one side of the equation, apply it to the other as well to maintain balance.

The result leaves \( t \) isolated, which helps us determine its value.

Simple Linear Equations

Linear equations, like the one here \( 7t = 42 \), are some of the most straightforward equations you can encounter. These equations graph as straight lines and reflect a constant rate of change.
Here’s what characterizes them:

• They typically appear in the form \( ax + b = c \), though constants \( b\) or \( c \) can sometimes be zero.
• In our example, the equation lacks a constant \( b \), making it a bit simpler: \( 7t = 42 \).
• A significant trait of simple linear equations is they have exactly one solution. This means there is a specific, single value that \( t \) can take to satisfy the equation.

To solve these, follow straightforward steps like those outlined above, emphasizing operations like addition, subtraction, multiplication, or division with a focus on opposites to isolate the variable.

Division in Equations

Division plays a crucial role in solving equations, especially when isolating a variable. Consider our equation \( 7t = 42 \).
Here's how division helps:

• When a variable is multiplied by a number, such as \( 7t \), division "undoes" this multiplication. To "free" \( t \) from the multiplication by 7, we divide both sides of the equation by 7.
• This operation simplifies \( \frac{7t}{7} = \frac{42}{7} \), effectively reducing to \( t = 6 \). It's important to divide both sides to keep the equation balanced.
• Maintaining balance means whatever you do to one side, do to the other, ensuring the equation remains equivalent.

By applying division correctly, you reverse the effects of multiplication, providing the desired outcome: an isolated variable.