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Chapter 2: Problem 56

Solve each equation, and check your solution. $$ 4 x-3-8 x+1=-5 x+9 $$

Short Answer

Expert verified
x = 11

Step by step solution

01

Combine like terms

First, combine the terms with x on the left side of the equation: \[ 4x - 8x - 3 + 1 = -5x + 9 \] This simplifies to: \[ -4x - 2 = -5x + 9 \]
02

Move x terms to one side

Add 5x to both sides to move the x terms to one side of the equation: \[ -4x + 5x - 2 = 9 \] This simplifies to: \[ x - 2 = 9 \]
03

Isolate x

Add 2 to both sides to isolate x: \[ x - 2 + 2 = 9 + 2 \] This simplifies to: \[ x = 11 \]
04

Check the solution

Substitute x = 11 back into the original equation to check the solution: \[ 4(11) - 3 - 8(11) + 1 = -5(11) + 9 \] Simplify both sides: \[ 44 - 3 - 88 + 1 = -55 + 9 \] \[ -46 = -46 \] 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.

Combining Like Terms
When solving linear equations, one of the first steps is to simplify the equation by combining like terms.
Like terms are terms that contain the same variables raised to the same power. They can be combined by adding or subtracting their coefficients.

For example, in the equation ewline [4x - 3 - 8x + 1 = -5x + 9ewline], we can combine the terms with 'x' and the constant terms separately:

ewline [4x - 8xewline] simplifies to ewline [-4xewline], and ewline [-3 + 1ewline] simplifies to ewline [-2ewline]. This gives us:ewline [-4x - 2 = -5x + 9ewline]Combining like terms helps to reduce the equation to a simpler form, making it easier to solve.
Isolating Variables
The next crucial step is to isolate the variable. This means getting the variable on one side of the equation and everything else on the other side.
In our example, we want to isolate 'x'. After combining like terms, our equation is ewline [-4x - 2 = -5x + 9ewline].

To isolate 'x', we need to get rid of the 'x' terms on one side. We do this by adding 5x to both sides: ewline [-4x + 5x - 2 = 9ewline], which simplifies to ewline [x - 2 = 9ewline].

Now, to completely isolate 'x', we add 2 to both sides:ewline [x - 2 + 2 = 9 + 2ewline], which simplifies to ewline [x = 11ewline].

At this stage, 'x' is isolated, and we've solved the equation.
Checking Solutions
After solving the equation, it's important to check the solution to ensure it's correct. This step verifies the accuracy of your answer.
To check the solution, substitute the value of the variable back into the original equation and simplify both sides.

In our example, we found that x = 11. We substitute it back into the original equation:ewline [4(11) - 3 - 8(11) + 1 = -5(11) + 9ewline].

Simplify both sides:ewline [44 - 3 - 88 + 1 = -55 + 9ewline].
This simplifies to ewline [-46 = -46ewline], which is true.

Since both sides of the equation are equal, our solution x = 11 is correct.

Always remember to check your solutions to avoid mistakes and ensure your final answer is accurate.

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