Problem 30 - Elementary Algebra

Question

Solve. $$3 x+2-17 x-2=-48 \times 2-4$$

Step-by-Step Solution

Answer

The solution is \(x = \frac{50}{7}\).

Step 1: Simplify the Equation

Start by simplifying both sides of the equation. Combine like terms in the expression on the left. For the left-hand side, combine the terms with \(x\): \(3x - 17x = -14x\), and the constants \(2 - 2 = 0\). So the left-hand side becomes \(-14x\). In the right-hand side, compute the multiplication and subtraction: \(-48 \times 2 - 4 = -96 - 4 = -100\). The equation now looks like \(-14x = -100\).

Step 2: Isolate the Variable

To solve for \(x\), divide both sides of the equation by \(-14\) to isolate \(x\). This gives \(x = \frac{-100}{-14}\).

Step 3: Simplify the Fraction

Simplify the fraction \(\frac{-100}{-14}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This results in \(x = \frac{100}{14} = \frac{50}{7}\).

Key Concepts

Combining Like Terms

Combining like terms is a crucial step in solving linear equations. It's like cleaning up your room before you start doing your homework. When solving equations, you'll often come across similar variables or constants that can be combined for simplicity. Like terms are terms that have the same variable raised to the same power. For example, in the expression \(3x - 17x + 2 - 2\), both \(3x\) and \(-17x\) are like terms because they both contain the variable \(x\).

• Why Combine Like Terms? By combining like terms, you simplify the equation, which makes it easier to solve.
• Example: Combine \(3x\) and \(-17x\) to get \(-14x\).
• Constants: Also combine any constants that are on the same side of the equation, like \(2\) and \(-2\), which in this case cancel each other out to become \(0\).

Simplification reduces the equation to fewer terms, and in this exercise, it results in \(-14x = -100\) which sets you up to solve for \(x\).

Isolating the Variable

Isolating the variable is all about getting the unknown by itself on one side of the equation. This is like focusing on one single challenge at a time. Once you've cleaned up the equation by combining like terms, the next step is to solve for the variable that you are looking for, often \(x\).

• Get it Alone: You want to perform operations that will leave \(x\) alone on one side of the equation.
• Example: In \(-14x = -100\), divide both sides by \(-14\) to get \(x\) by itself. This results in \(x = \frac{-100}{-14}\).
• Operations: The operations you can use include addition, subtraction, multiplication, and division—whichever applies to eliminate the numbers attached to the variable.

Remember that whatever operation you perform on one side, you must also perform on the other to keep the equation balanced.

Simplifying Fractions

Once you've isolated the variable, you might end up with a fraction. Don't be intimidated! Simplifying fractions is like condensing your notes. It might seem complex at first, but it's simply a matter of making things cleaner and more understandable.

• What is Simplifying? This means making the fraction as simple as possible by ensuring the numerator and denominator have no common factors other than 1.
• Example: Look at \(\frac{-100}{-14}\). Both the numerator and the denominator can be divided by 2, reducing the fraction to \(\frac{50}{7}\).
• Simplification: This process helps to present the solution in its simplest form, which is often easier to work with and understand.

Simplifying fractions ensures that your final answer is not only correct but also neat and precise.