Problem 64 - Elementary Algebra

Question

Solve. $$ 13(x-2)+15=19(3 x+3) $$

Step-by-Step Solution

Answer

The solution is \(x = -\frac{17}{11}\).

Step 1: Distribute the Numbers

Start by distributing the numbers outside the parentheses to the terms inside the parentheses on both sides of the equation. For the left side, distribute 13 into \((x - 2)\), leading to: \(13x - 26\). For the right side, distribute 19 into \((3x + 3)\), which gives: \(57x + 57\). Now rewrite the equation: \[ 13x - 26 + 15 = 57x + 57 \]

Step 2: Combine Like Terms

Simplify both sides of the equation by combining like terms. On the left side, combine \(-26\) and \(15\), resulting in: \(-11\). Thus, the equation becomes: \[ 13x - 11 = 57x + 57 \]

Step 3: Move Variables to One Side

To solve for \(x\), you'll want all the \(x\) terms on one side of the equation. Subtract \(13x\) from both sides: \[ -11 = 44x + 57 \]

Step 4: Move Constants to the Other Side

Isolate the \(x\) term by moving the constant number from one side to the other. Subtract \(57\) from both sides: \[ -68 = 44x \]

Step 5: Solve for x

To solve for \(x\), divide both sides of the equation by \(44\): \[ x = \frac{-68}{44} \] Simplify the fraction: \[ x = -\frac{17}{11} \]

Key Concepts

Distributive Property

In algebra, the distributive property is a fundamental technique that helps simplify expressions. It allows us to eliminate parentheses by distributing the multiplication over addition or subtraction. In the exercise, we apply this property to clear the parentheses on both sides of the equation.

• On the left side: Multiply 13 with each term inside \( (x - 2) \) resulting in \(13x - 26\).
• On the right side: Multiply 19 with each term in \( (3x + 3) \) giving \(57x + 57\).

Once these calculations are done, the equation will have no more parentheses, making it easier to combine terms and solve it. Understanding the distributive property is important as it simplifies complex algebraic equations into manageable forms for further steps.

Combining Like Terms

Combining like terms is another crucial step in simplifying algebraic expressions. This process involves combining terms that have the same variable raised to the same power. This simplification helps reduce the equation, making it easier to solve.In our exercise:

• On the left side: Combine constant terms \(-26\) and \(+15\) to get \(-11\). This step simplifies the equation without changing the variable terms, turning the expression into \(13x - 11\).

By combined like terms, we effectively reduce the complexity of the equation. This step sets the scene for "isolating the variable," the last part needed to find the solution for the variable, \(x\).

Isolating the Variable

The final goal of solving an algebraic equation is to isolate the variable, which here is \(x\). Isolating \(x\) means rearranging the equation so that \(x\) stands alone on one side, showing what it equals.In this exercise:

• Subtract \(13x\) from both sides to collect all \(x\) terms on one side. This transforms the equation into \(-11 = 44x + 57\).
• Then, subtract 57 from both sides to move constants to the opposite side of \(x\), which changes the equation to \(-68 = 44x\).
• Finally, divide both sides by 44 to completely isolate \(x\). This results in \(x = \frac{-68}{44}\), which simplifies to \(x = -\frac{17}{11}\).

At this step, we achieve a clear solution for \(x\). Isolating the variable is a powerful method in algebra, helping reveal the value of unknowns clearly and concisely.