Problem 32 Solve each equation, and check y... [FREE SOLUTION]

91影视

Beginning Algebra

Margaret L. Lial, John Hornsby

$Math Studyset 91影视 Explanations$ Math

11 Edition

Chapter 2: Problem 32

Solve each equation, and check your solution. $$ 8 t+5=7 t $$

Short Answer

Expert verified

The solution is $ t = -5 $.

Step by step solution

- Move Variable Terms to One Side

Subtract $ 7t $ from both sides to consolidate the variable terms on one side of the equation. \[ 8t + 5 - 7t = 7t - 7t \] which simplifies to: \[ t + 5 = 0 \]

- Isolate the Variable

Subtract 5 from both sides to isolate $ t $. \[ t + 5 - 5 = 0 - 5 \] which simplifies to: \[ t = -5 \]

- Check the Solution

Substitute $ t = -5 $ back into the original equation to verify. \[ 8(-5) + 5 = 7(-5) \] Simplify both sides: \[ -40 + 5 = -35 \] \[ -35 = -35 \] Since both sides of the equation are equal, the solution is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Equation Solving

Solving a linear equation involves finding the value of the variable that makes the equation true. Linear equations are equations of the first degree, meaning they have variables raised only to the first power. In our example, we start with the equation:
8t + 5 = 7t
The goal is to manipulate this equation such that we isolate the variable (t in this case) on one side of the equation. By systematically simplifying the expressions, we can solve for the unknown variable. Getting to the solution involves understanding basic arithmetic operations such as addition, subtraction, multiplication, and division.

Variable Isolation

Variable isolation is a crucial step in solving linear equations. This means rearranging the equation until we get the variable alone on one side:
For our example:
Step 1: Move Variable Terms to One Side
Subtract 7t from both sides of the original equation to consolidate the terms:
8t + 5 - 7t = 7t - 7t
This simplifies to:
t + 5 = 0
Step 2: Isolate the Variable
Subtract 5 from both sides:
t + 5 - 5 = 0 - 5
Which simplifies to:t = -5
By carefully managing each operation, you isolate the variable (t) and find its value.

Solution Verification

It鈥檚 essential to verify your solution to ensure that it is correct. This involves substituting the found value back into the original equation:
In our solved equation, we found t = -5.
We substitute -5 for t in the original equation:
8(-5) + 5 = 7(-5)
Simplify both sides:
-40 + 5 = -35
-35 = -35
Since both sides of the equation are equal, the solution is verified as correct. This step confirms that our algebraic manipulations and result are accurate. Always verify your solutions to avoid mistakes and ensure understanding.

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91影视

Solve each equation, and check your solution. $$ 8 t+5=7 t $$

Short Answer

Step by step solution

- Move Variable Terms to One Side

- Isolate the Variable

- Check the Solution

Key Concepts

Equation Solving

Variable Isolation

Solution Verification

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