Problem 27 - College Algebra with Modeling and Visualization

Question

Solve the equation and check your answer. $$ -4(5 x-1)=8-(x+2) $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = -\frac{2}{19}\).

Step 1: Distribute Constants

Distribute the constants next to the parentheses for both sides of the equation. The left side: \(-4(5x - 1) = -4 \cdot 5x + (-4) \cdot (-1) = -20x + 4\).The right side:\(8 - (x + 2) = 8 - x - 2\).

Step 2: Simplify Both Sides

Simplify the expressions from Step 1:- Left side: \(-20x + 4\).- Right side: Combine like terms: \(8 - 2 - x = 6 - x\).

Step 3: Set Simplified Equation

Set the simplified expressions equal to each other:\(-20x + 4 = 6 - x\).

Step 4: Isolate Variable Terms

Add \(x\) to both sides to move all \(x\) terms to one side:\(-20x + x + 4 = 6\).This simplifies to:\(-19x + 4 = 6\).

Step 5: Isolate Constant Terms

Subtract 4 from both sides to move constants to the other side:\(-19x = 6 - 4\).Simplifying gives: \(-19x = 2\).

Step 6: Solve for x

Divide both sides by \(-19\) to solve for \(x\):\(x = \frac{2}{-19}\), which simplifies to:\(x = -\frac{2}{19}\).

Step 7: Check the Solution

Substitute \(x = -\frac{2}{19}\) back into the original equation to verify.Starting with:\(-4(5(-\frac{2}{19}) - 1) = 8 - (-\frac{2}{19} + 2)\).Calculate each step:- Left side: \(-4(-\frac{10}{19} - 1) = -4(-\frac{10}{19} - \frac{19}{19}) = -4(-\frac{29}{19}) = \frac{116}{19}\).- Right side: \(8 - (\frac{-2}{19} + \frac{38}{19}) = 8 - (\frac{36}{19}) = \frac{152}{19} - \frac{36}{19} = \frac{116}{19}\).Since both sides equal \(\frac{116}{19}\), the solution is verified.

Key Concepts

Distributive Property

The distributive property is a foundation of algebra that allows us to multiply a single term across terms inside parentheses. This principle can be represented as: \( a(b + c) = ab + ac \). It means that you distribute or "hand out" the multiplication across the terms inside the parentheses. In solving our equation \(-4(5x - 1)\), you first distribute \(-4\) to each term inside the parentheses.

• First, multiply \(-4\) by \(5x\) to get \(-20x\).
• Then, multiply \(-4\) by \(-1\) to get \(+4\).

Putting these together, the left side simplifies to \(-20x + 4\).
The distributive property helps to break down and simplify problems into more manageable parts, making it easier to combine like terms later.

Combining Like Terms

Combining like terms is a method used to simplify algebraic expressions. Like terms are terms in an equation that have the same variable parts, meaning the same variables raised to the same power. By adding or subtracting these terms, you can simplify an expression.For example, once we have expanded \(-4(5x - 1)\) to \(-20x + 4\), we look at the right side of the original equation, which was simplified to \(6 - x\).

• Recognize that "\(6\)" is a constant and "\(-x\)" is a term with a variable.
• When combing like terms, arrange all constants together and all variables together.

Simplifying allows you to set the equation as \(-20x + 4 = 6 - x\), where you can straightforwardly identify all similar components. This sets the stage for isolating the variable terms on one side.

Solving Linear Equations

Solving linear equations involves finding the value of the variable that makes the equation true. After simplifying the initial equation with the distributive property and combining like terms, we progress to solving for \(x\).To isolate the variable, follow these general steps:

• Move variable terms to one side: In the equation \(-20x + 4 = 6 - x\), add \(x\) to both sides to bring all \(x\) terms together, resulting in \(-19x + 4 = 6\).
• Isolate constant terms: Subtract 4 from both sides to move constant terms away from the variable terms, simplifying to \(-19x = 2\).
• Divide to solve for the variable: Divide both sides by \(-19\) to solve for \(x\), which gives \(x = -\frac{2}{19}\).

Lastly, always verify your solution by substituting it back into the original equation. This ensures that both sides of the equation balance out, confirming the accuracy of your solution. By checking the solution, you can be confident that \(x = -\frac{2}{19}\) is correct.