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Chapter 5: Problem 13

Solve and check. $$4-3 w=-2$$

Short Answer

Expert verified
The solution for \( w \) is \( 2 \). This was verified by substituting \( w = 2 \) back into the original equation.

Step by step solution

01

Isolate the variable

To isolate \( w \), first subtract '4' from both sides of the equation, which gives \( -3w = -2 - 4 \) or \( -3w = -6 \). The minus sign on \( w \) needs to be removed in order to isolate \( w \).
02

Simplify the equation

To remove the negative sign from \( -3w \), divide both sides of the equation by '-3', yielding \( w = -6 / -3 \). This gives \( w = 2 \).
03

Check the solution

Lastly, we insert \( w = 2 \) back into the original equation to verify that the solution is correct. \( 4 - 3(2) = 4 - 6 = -2 \). This checks out, as the left side is equal to the right side, verifying that \( w = 2 \) is indeed the correct solution.

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Key Concepts

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

Isolate the Variable
The first step in solving linear equations is to isolate the variable. This means manipulating the equation to have the variable on one side and all the constants on the other. Let's take the equation from the example, 4 - 3w = -2.

The goal is to move all terms without the variable to the other side of the equality. This is done through operations that maintain the equality, such as adding or subtracting the same amount from both sides. In our case, subtracting '4' from both sides allows the variable term to be on one side: -3w = -6. Isolation is a crucial step, giving us a clear view of what operation is needed to solve for the variable.
Simplifying Equations
Once the variable is isolated, the next objective is simplifying the equation. This involves performing arithmetic operations to solve for the variable. In the given equation, we are left with -3w = -6 after isolating the variable. To simplify, observe the coefficient of the variable, which is '-3' in our case.

By dividing both sides of the equation by this coefficient, we eliminate the coefficient, leaving the variable by itself: w = 2. Simplification is all about making the equation easier to read and solve, which often spells the difference between finding the correct solution quickly and making errors.
Checking Solutions
Finally, after simplifying the equation and finding a value for the variable, it's essential to verify the solution. By substituting the obtained value back into the original equation, we confirm its validity.

Checking is a critical step that prevents the acceptance of incorrect solutions due to errors in calculation. For the exercise 4 - 3w = -2, inserting our solution w = 2 back in gives us 4 - 3(2) = 4 - 6 = -2, confirming that the left side does indeed equal the right, and our solution is correct. Always remember, a solution isn鈥檛 truly a solution until it's been checked and verified.

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