Problem 46 - Elementary Algebra

Question

Solve. $$ -5(2 x-1)+3=-12 $$

Step-by-Step Solution

Answer

The solution to the equation is \( x = 2 \).

Step 1: Expand the Equation

Firstly, distribute the -5 throughout the terms inside the parentheses. This means you multiply -5 with each term inside: \[-5(2x) - 5(-1) + 3 = -12\] This simplifies to:\[-10x + 5 + 3 = -12\].

Step 2: Combine Like Terms

Next, combine any like terms on the left side of the equation. Here, the constant terms 5 and 3 can be combined:\[-10x + 8 = -12\].

Step 3: Isolate the x-term

Subtract 8 from both sides of the equation to isolate the x-related term:\[-10x = -12 - 8\]. This simplifies to:\[-10x = -20\].

Step 4: Solve for x

Finally, divide both sides by -10 to solve for x:\[x = \frac{-20}{-10}\]. After performing the division, this results in:\[x = 2\].

Key Concepts

Equation Solving

Equation solving is the process of finding values for variables that make the equation true. In the problem at hand, we need to find the value of \(x\) that satisfies the equation \[-5(2x-1)+3=-12\].To solve it efficiently, we follow a few strategic steps:

• Expand the expression: We employ the distributive property to remove parentheses and simplify the equation.
• Combine like terms: Add or subtract constants to simplify further.
• Isolate the variable: Use operations to get the variable to one side of the equation.
• Solve for the variable: Complete the process to find the variable's value.

Understanding these steps makes solving equations systematic and reduces the chance of errors.

Linear Equations

Linear equations are mathematical expressions where the highest power of the variable is one. They appear as straight lines when graphed. An example of a simple linear equation is \(ax + b = c\).The equation in our exercise, \(-5(2x-1)+3=-12\), is linear because:

• The variable \(x\) has an exponent of one.
• There are no terms like \(x^2\) or higher powers.
• It involves basic operations like addition, subtraction, and multiplication.

Linear equations are fundamental in algebra. Solving them often involves straightforward operations, allowing students to develop skills that will be valuable for more complex topics.

Distributive Property

The distributive property is a key algebraic property used to simplify and solve equations. It involves distributing a multiplied term across a sum or difference within parentheses. This property is expressed as \(a(b + c) = ab + ac\).In our problem, the term \(-5(2x - 1)\) needs to be expanded using the distributive property:

• Multiply \(-5\) by \(2x\) to get \(-10x\).
• Multiply \(-5\) by \(-1\) to get \(+5\).

This operation is vital for removing parentheses, making the equation easier to manipulate. By mastering the distributive property, students can tackle a wide range of algebraic problems efficiently.