Problem 26 - College Algebra with Modeling and Visualization

Question

Solve the equation and check your answer. $$ -3(5-x)-(x-2)=7 x-2 $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{-11}{5} \).

Step 1: Distribute and Simplify

Begin by distributing the terms on the left side of the equation: \[ -3(5-x) -(x-2) = -15 + 3x - x + 2 \]Combine like terms:\[ 2x - 13 \] So the equation becomes:\[ 2x - 13 = 7x - 2 \]

Step 2: Move Variables to One Side

Subtract \(2x\) from both sides to begin isolating the variable \(x\):\[ -13 = 7x - 2x - 2 \]This simplifies to:\[ -13 = 5x - 2 \]

Step 3: Isolate the Variable Term

Add 2 to both sides to isolate the \(5x\) term:\[ -13 + 2 = 5x \]This simplifies to:\[ -11 = 5x \]

Step 4: Solve for x

Divide each side by 5 to solve for \(x\):\[ x = \frac{-11}{5} \]

Step 5: Check the Solution

Substitute \(x = \frac{-11}{5}\) back into the original equation to verify the solution:- Compute the left side: \[ -3\left(5 - \frac{-11}{5}\right) - \left(\frac{-11}{5} - 2\right) \] Simplify each part: \[ -3\left(\frac{25 + 11}{5}\right) - \left(\frac{-11 - 10}{5}\right) \] \[ -3\cdot\frac{36}{5} + \frac{21}{5} \] \[ -\frac{108}{5} + \frac{21}{5} = \frac{-87}{5} \]- Compute the right side: \[ 7 \cdot \frac{-11}{5} - 2 \] \[ \frac{-77}{5} - \frac{10}{5} = \frac{-87}{5} \]Both sides equal, confirming the solution is correct.

Key Concepts

Distributive Property

When solving equations, the distributive property is a valuable tool. It's used to eliminate parentheses by multiplying a single term outside the parenthesis by each term inside it. This allows us to simplify expressions and make them easier to work with. For example, in the equation \( -3(5-x)-(x-2) = 7x-2 \), we apply the distributive property by multiplying \(-3\) by both \(5\) and \(-x\), which gives us \(-15\) and \(3x\).

• The minus sign in front of \((x-2)\) impacts both terms, turning it into \(-x+2\).
• This step is crucial for breaking down complex expressions into simpler parts.

The result of distribution is \(-15 + 3x - x + 2\), and it prepares us for the next steps in solving the equation.

Combining Like Terms

Once we've used the distributive property to expand the terms, the next step in our process is combining like terms. "Like terms" are terms in an expression that have identical variable parts, although they may have different coefficients. In our example, \(-15 + 3x - x + 2\), we can combine \(3x - x\) as these are both terms involving the variable \(x\). This simplifies to \(2x\).

• Similarly, the constant terms \(-15\) and \(+2\) can be combined, becoming \(-13\).
• The new expression becomes \(2x - 13\).

Combining like terms helps to consolidate the equation, making it more manageable to work with in further steps.

Isolation of Variables

The isolation of variables is a pivotal step in solving equations, especially when solving for an unknown variable like \(x\). The goal is to get \(x\) by itself on one side of the equation. Starting from \(2x - 13 = 7x - 2\), we move terms around to isolate \(x\). Subtract \(2x\) from both sides, resulting in \(-13 = 5x - 2\). Following this, we add \(2\) to both sides to clear out the constant term next to the variable term, giving \(-11 = 5x\).

• Through these manipulations, we progressively center \(x\) on one side.
• The final step is to solve for \(x\) by dividing both sides by \(5\), leading to \(x = \frac{-11}{5}\).

Isolating the variable is essential in giving us the value of \(x\). By performing inverse operations step by step, we can isolate \(x\) efficiently.

Checking Solutions

After finding a solution to an equation, checking that solution is vital to ensure its correctness. To do this, substitute the found value back into the original equation and see if both sides remain equal. Begin by substituting \(x = \frac{-11}{5}\) into \(-3(5-x)-(x-2)=7x-2\). Calculate each side separately.

• The left side simplifies to \(\frac{-87}{5}\) after computing \(-3\left(\frac{36}{5}\right)+\frac{21}{5}\).
• The right side also equals \(\frac{-87}{5}\) upon evaluating \(7 \cdot \frac{-11}{5} - 2\).

Since both sides are equal, the solution \(x = \frac{-11}{5}\) is verified. Checking solutions is a step that provides confidence that the calculations and transformations were done correctly.