Problem 43 - Algebra for College Students

Question

Solve each equation. \(3(x-4)-7(x+2)=-2(x+18)\)

Step-by-Step Solution

Answer

The solution is \(x = 5\).

Step 1: Expand the expressions

Distribute the numbers outside the parentheses into the expressions inside. For the left side: \[3(x-4) - 7(x+2) = 3x - 12 - 7x - 14\]For the right side: \[-2(x+18) = -2x - 36\] So, the equation simplifies to: \[3x - 12 - 7x - 14 = -2x - 36\]

Step 2: Combine Like Terms

Combine the like terms on the left side of the equation.Combine terms involving \(x\): \[3x - 7x = -4x\]Combine the constant terms: \[-12 - 14 = -26\]This gives us: \[-4x - 26 = -2x - 36\]

Step 3: Move Variable Terms to One Side

To move all the variable terms to one side, add \(4x\) to both sides:\[-4x - 26 + 4x = -2x - 36 + 4x\]This simplifies to:\[-26 = 2x - 36\]

Step 4: Isolate the Variable Term

Add 36 to both sides to isolate the variable term:\[-26 + 36 = 2x - 36 + 36\]This results in:\[10 = 2x\]

Step 5: Solve for x

Divide both sides by 2 to solve for \(x\):\[\frac{10}{2} = \frac{2x}{2}\]This simplifies to:\[x = 5\]

Key Concepts

Distributive Property

The distributive property is a useful tool when solving linear equations as it helps simplify expressions that have parentheses. When you encounter an equation like \(3(x-4)\), the distributive property allows you to multiply the 3 with each term inside the parentheses, meaning you'll perform two different multiplications. Here's how it looks:

• Multiply 3 with \(x\) to get \(3x\).
• Multiply 3 with \(-4\) to get \(-12\).

Applying the distributive property helps reduce complex expressions into simpler terms, making it easier to work through the problem. In the problem given above, the same rule applies to other terms:

• \(-7(x+2)\) becomes \(-7x - 14\).
• \(-2(x+18)\) becomes \(-2x - 36\).

By using the distributive property effectively, you can turn expressions with parentheses into linear expressions that are easier to manage, paving the way for solving the equation.

Combining Like Terms

After using the distributive property, the next step is combining like terms. Like terms are terms that have the same variables raised to the same power. By grouping them together, we simplify the equation further. Let's see how it works:

• From \(3x - 7x\), you combine to get \(-4x\).
• For the constants \(-12 - 14\), you add them to get \(-26\).

Combining these like terms makes the equation less cluttered, which simplifies your work and helps to see clearly what steps to take next. This step ensures that all similar terms are made into single terms, making it easier to resolve the later steps when isolating variables.

Isolating Variables

After combining like terms, the goal is to isolate the variable on one side of the equation. This sets the stage for solving the equation. Here’s how to isolate the variable:

• Once you have \(-4x - 26 = -2x - 36\), decide which variable term you want to move. Adding \(4x\) to each side results in \(-26 = 2x - 36\).
• Add \(36\) to both sides to have only the variable term on one side: \(-26 + 36 = 2x\).
• This results in \(10 = 2x\).

Isolating the variable means getting all terms with the variable on one side and the constants on the opposite side. This step is crucial for simplifying equations to a form where solving for the variable becomes straightforward.

Steps in Solving Equations

Solving equations requires a clear, sequential approach. Each step taken helps simplify the problem further, eventually leading to the solution. Here are the key steps used in solving the linear equation in this task:

• Distribute: Apply the distributive property to remove parentheses.
• Combine: Combine like terms to simplify the equation.
• Isolate Variable: Move all variable terms to one side and constants to the other.
• Solve: Finally, solve for the variable by performing basic operations like addition, subtraction, multiplication, or division on both sides.

This systematic approach ensures that every part of the equation is broken down into manageable sections. By following these steps methodically, solving an equation becomes an organized process, leading you directly to the correct solution, like in the example where we found that \(x = 5\).