Problem 53 - Algebra A Combined Function

Question

Solve. $$ -6.5-4 x-1.6-3 x=-6 x+9.8 $$

Step-by-Step Solution

Answer

The solution is \(x = -17.9\).

Step 1: Rewrite the Equation

Begin by rewriting the original equation clearly:\[-6.5 - 4x - 1.6 - 3x = -6x + 9.8\]

Step 2: Combine Like Terms on One Side

Combine the like terms on the left side of the equation:\[-6.5 - 1.6 - 4x - 3x = -6x + 9.8\]Simplify:\[-8.1 - 7x = -6x + 9.8\]

Step 3: Move Variable Terms to One Side

Add \(7x\) to both sides to get all variable terms on one side:\[-8.1 = x + 9.8\]

Step 4: Isolate the Variable

Subtract 9.8 from both sides to solve for \(x\):\[-8.1 - 9.8 = x\]Calculate:\[x = -17.9\]

Step 5: Verify the Solution

Substitute \(x = -17.9\) back into the original equation to verify the solution:\[-6.5 - 4(-17.9) - 1.6 - 3(-17.9) = -6(-17.9) + 9.8\]Calculate both sides to verify that they are equal.

Key Concepts

Combining Like Terms

One of the essential skills in solving linear equations is combining like terms. It simplifies equations by grouping terms that share the same variable or are constant.
For example, in the equation \[ -6.5 - 4x - 1.6 - 3x = -6x + 9.8 \] the like terms on the left side are the numerical constants \[-6.5\] and \[-1.6\], as well as the terms involving \(x\): \(-4x\) and \(-3x\).

• First, add the constant numbers together:\[-6.5 - 1.6 = -8.1\].
• Next, combine the terms that contain the variable \(x\): \(-4x - 3x = -7x\).

Together, this reformed equation becomes \[-8.1 - 7x = -6x + 9.8\]. Notice how the expression becomes more straightforward and manageable.

Isolating Variables

Isolating the variable is a crucial method used to solve for the unknown in an equation. Once you've combined like terms, the next step is to get the variable on one side of the equation.
In the equation \[-8.1 - 7x = -6x + 9.8\], we want to keep all the \(x\) terms on one side. By adding \(7x\) to both sides, we eliminate the variable from the left side:

• This results in: \[-8.1 = x + 9.8\].
• Afterwards, subtract \(9.8\) from both sides to solve for \(x\): \[-8.1 - 9.8 = x\].

Now it's simply a matter of calculation.

Substitution Method

The substitution method involves replacing the variable with its calculated value to ensure that the solution is correct. After solving for \(x\), it's essential to substitute it back into the original equation.
Given \(x = -17.9\), substitute it into \[-6.5 - 4x - 1.6 - 3x = -6x + 9.8\].

• Replace each instance of \(x\) with \(-17.9\): \[-6.5 - 4(-17.9) - 1.6 - 3(-17.9) = -6(-17.9) + 9.8\].

By calculating both sides, we verify if they equal each other, confirming the solution is correct.

Mathematical Verification

Verification is the final check to confirm the accuracy of your solution. It ensures that the value found for the variable makes both sides of the original equation equal.
After substituting \(x = -17.9\), compute both sides:

• Calculate the left side: \[-6.5 + 71.6 - 1.6 + 53.7 = 117.2\].
• Calculate the right side: \[107.4 + 9.8 = 117.2\].
• Since both sides equal 117.2, the solution \(x = -17.9\) is verified.

Verification is crucial as it highlights errors if both sides do not match, prompting another look at your calculations.