Problem 76 - Algebra A Combined Function

Question

\(-19+74=-5(x+3)\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = -14\).

Step 1: Simplify the Left Side

First, simplify the left side of the equation by performing the arithmetic operation. We have the equation: \(-19 + 74\).Calculate this:\(74 - 19 = 55\).So, the left side simplifies to 55. The equation now looks like: \(55 = -5(x+3)\).

Step 2: Distribute on the Right Side

On the right side, distribute the \(-5\) into the parenthesis:\(-5(x + 3) = -5x - 15\).Replace the right side of the equation with the expanded expression:\(55 = -5x - 15\).

Step 3: Add 15 to Both Sides

To isolate terms with \(x\), first add 15 to both sides of the equation:\(55 + 15 = -5x - 15 + 15\).Simplify both sides:\(70 = -5x\).

Step 4: Solve for x

Now, solve for \(x\) by dividing both sides by \(-5\):\(70 \div (-5) = -5x \div (-5)\).Simplify to find \(x\):\(x = -14\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to simplify and solve equations effectively. At its core, this property states that multiplying a number by a sum is the same as multiplying each addend by the number and then adding the products. Let's break it down with the equation from the exercise.
When dealing with \(-5(x+3)\), you apply the distributive property by multiplying \(-5\) with both \(x\) and \(3\), leading to \(-5x - 15\).

• Multiply \(-5\) by \(x\): results in \(-5x\).
• Multiply \(-5\) by \(3\): results in \(-15\).

This step ensures that you have correctly expanded the equation, which is crucial for further steps in solving linear equations. Understanding and applying the distributive property correctly allows one to "break out" terms within parentheses, streamlining the path to finding the variable.

Solving Equations

Solving linear equations is all about finding the value of the unknown variable that makes the equation true. This involves performing operations in a systematic manner to keep both sides of the equation balanced. Our ultimate goal is to isolate the variable, making it the centerpiece of our solution.
To solve the equation \(55 = -5x - 15\), you must first simplify and then perform operations that eliminate constants and coefficients.

• Add \(15\) to both sides: This cancels out the \(-15\) on the right side, giving you \(70 = -5x\).
• Divide both sides by \(-5\): \(70 \div (-5) = x\). This division will cancel out the \(-5\) coefficient of \(x\), isolating it as \(x = -14\).

During each step, consistency is key. Keeping the equation balanced by mirroring operations on both sides ensures correctness and leads you step by step towards the solution.

Isolation of Variables

Isolation of variables is the process of rearranging an equation to get the unknown variable on one side, ideally by itself. In algebra, this is a primary technique used to solve for the variable's specific value.
Initially, our equation was: \(55 = -5x - 15\).

• First step: Add \(15\) to both sides, leading to \(70 = -5x\).
• Second step: Divide both sides by \(-5\), resulting in \(x = -14\).

By isolating \(x\), we focused on transforming the equation step-by-step, ensuring each operation gradually led to the intended outcome. This methodical approach clears other terms and makes the variable the subject. Isolation of variables not only aids in solving equations but also enhances understanding of mathematical relationships and structures.