Problem 30 - Algebra for College Students

Question

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction. \(5(x+3)+4 x-5=4-2 x\)

Step-by-Step Solution

Answer

The solution is \( x = \frac{-6}{11} \). This equation is neither an identity nor a contradiction.

Step 1: Distribute the terms

First, distribute the 5 to both terms inside the parentheses: \[ 5(x+3) + 4x - 5 = 4 - 2x \] This becomes: \[ 5x + 15 + 4x - 5 = 4 - 2x \]

Step 2: Combine like terms

Combine the like terms on the left side of the equation: \[ 5x + 4x + 15 - 5 = 4 - 2x \] This simplifies to: \[ 9x + 10 = 4 - 2x \]

Step 3: Move variable terms to one side

Add 2x to both sides to move all variable terms to one side: \[ 9x + 2x + 10 = 4 \] This gives us: \[ 11x + 10 = 4 \]

Step 4: Isolate the variable term

Subtract 10 from both sides to isolate the variable term: \[ 11x + 10 - 10 = 4 - 10 \] Now we have: \[ 11x = -6 \]

Step 5: Solve for the variable

Divide both sides by 11 to solve for x: \[ x = \frac{-6}{11} \]

Step 6: Check the solution

Substitute \( x = \frac{-6}{11} \) back into the original equation to check: \[ 5\left(\frac{-6}{11}+3\right) + 4\left(\frac{-6}{11}\right) - 5 = 4 - 2\left(\frac{-6}{11}\right) \] Simplifying both sides will show that the equation holds true, confirming the solution.

Key Concepts

distributive property

The distributive property is a fundamental concept in algebra. It allows us to break down expressions in parentheses and distribute the multiplication over addition or subtraction. In our example equation, we began with: \[ 5(x+3) + 4x - 5 = 4 - 2x \] Here, we used the distributive property to expand \(5(x+3)\). We multiply 5 by each term inside the parentheses: \[ 5 \times x + 5 \times 3 \] This simplifies to: \[ 5x + 15 \] After applying the distributive property, our equation looks like this: \[ 5x + 15 + 4x - 5 = 4 - 2x \] Now, we can move on to the next step.

combining like terms

Combining like terms means collecting terms that have the same variable to simplify the equation. Once we've distributed terms in our equation, we get: \[ 5x + 15 + 4x - 5 = 4 - 2x \] Look at the left side of the equation. We have two terms with \(x\): \[ 5x \text{ and } 4x \] Combine these: \[ 5x + 4x = 9x \] Next, we also need to combine the constant terms (numbers without a variable): \[ 15 - 5 = 10 \] So, our equation now simplifies to: \[ 9x + 10 = 4 - 2x \] Combining like terms helps us simplify and better visualize the equation. Now, we can move on to isolating the variable.

isolating the variable

Isolating the variable is a crucial step in solving linear equations. This means getting the variable on one side of the equation and everything else on the other. From our simplified equation: \[ 9x + 10 = 4 - 2x \] First, move \(-2x\) to the left by adding 2x to both sides of the equation: \[ 9x + 2x + 10 = 4 - 2x + 2x \] This combines to: \[ 11x + 10 = 4 \] Next, subtract 10 from both sides to move the constants away from the variable: \[ 11x + 10 - 10 = 4 - 10 \] Simplifying this gives us: \[ 11x = -6 \] Finally, divide both sides by 11 to solve for \(x\): \[ x = \frac{-6}{11} \] This is our solution. Always remember to check the solution by substituting it back into the original equation to ensure it satisfies the equation.