Problem 44 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each equation using both the addition and multiplication properties of equality. Check proposed solutions. $$-5 x=-2 x-12$$

Step-by-Step Solution

Answer

The solution of -5x = -2x - 12 is x = 4.

Step 1: Simplify the equation by using additive property of equality

Add 2x to both sides to maintain the balance of the equation. This results in an equation: -5x + 2x = -2x -12 + 2x. After simplifying this equation, it becomes: -3x = -12.

Step 2: Use multiplication property of equality to isolate 'x'

Divide both sides by -3 to isolate 'x'. The equation becomes: -3x/-3 = -12/-3. This simplifies to: x = 4.

Step 3: Check proposed solution

Substitute 'x' = 4 into the original equation to verify if the left side equals to the right side. -5*(4) equals to -2*(4) - 12. After performing the calculation, -20 equals to -20, therefore the solution 'x' = 4 is correct.

Key Concepts

Addition Property of Equality

In algebra, the addition property of equality is a fundamental concept that states if you add the same number to both sides of an equation, the equality is maintained. This is crucial for solving equations because it allows you to isolate variables and simplify expressions without changing the equation's overall balance.
For instance, if we start with the equation \(-5x = -2x - 12\), the goal is to get all terms involving \(x\) on one side. By adding \(2x\) to both sides, we use this property to maintain the balance, resulting in \(-5x + 2x = -2x + 2x - 12\).
Effectively, this means \(-3x = -12\), a simplified form that we can work on further. This property ensures that what we do on one side is mirrored on the other, crucial for maintaining the equation's integrity.

Multiplication Property of Equality

The multiplication property of equality is a second foundational concept in algebra. This property states that if you multiply both sides of an equation by the same non-zero number, the equation remains balanced. This property is incredibly useful, especially when trying to isolate a variable.
In our example, after using the addition property of equality, we were left with \(-3x = -12\). To solve for \(x\), we need \(x\) alone on one side. By dividing both sides by \(-3\) (which is equivalent to multiplying by the reciprocal), we rely on this property. Thus, the equation becomes \(-3x / -3 = -12 / -3\), simplifying to \(x = 4\).
This concept not only helps us to solve for variables but also demonstrates how multiplication or division can shift and shape equations while preserving their equality.

Equation Solving

Solving equations is a fundamental skill in algebra that involves finding the value of the unknown variable. This process often employs several algebraic properties, including both the addition and multiplication properties of equality.
Initially, in our exercise \(-5x = -2x - 12\), we used the addition property of equality to consolidate terms and simplify the equation. We then used the multiplication property to isolate the variable and solve for \(x\).
Finally, once a solution is proposed, it is critical to verify it by substituting back into the original equation. For \(x = 4\), substituting back results in a true statement: \(-5(4) = -2(4) - 12\), which simplifies to \(-20 = -20\). This verification step confirms the solution's accuracy, completing the equation solving process with confidence.