Problem 41 - Elementary Algebra

Question

Solve. $$ 7 x-5=-54 $$

Step-by-Step Solution

Answer

The solution is \(x = -7\).

Step 1: Add 5 to Both Sides

To isolate terms involving \(x\), we start by adding 5 to both sides of the equation. This gives us:\[7x - 5 + 5 = -54 + 5\]which simplifies to:\[7x = -49\]

Step 2: Divide by 7

Next, we divide both sides of the equation by 7 to solve for \(x\):\[\frac{7x}{7} = \frac{-49}{7}\]This simplifies to:\[x = -7\]

Key Concepts

Understanding Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. In algebra, we often use letters to represent numbers. This can help us form equations and solve various types of problems.

Algebra is like a language that allows us to describe mathematical ideas more generally. Think of it as a toolkit that lets us solve problems efficiently. When working with algebraic equations, you may often use:

• Variables: Letters like \(x\) or \(y\) that stand in for unknown numbers.
• Constants: Numbers that are always the same, like 5 or -2.
• Operations: Actions like addition, subtraction, multiplication, and division.

These components together allow us to formulate equations and solve for unknowns. Once you grasp the basics of algebra, it becomes much easier to handle more complex mathematical problems.

Mastering the Art of Solving Equations

Solving equations is a fundamental skill in algebra. It involves finding the value of a variable that makes an equation true. The key to solving equations is to perform operations that simplify the equation step by step until the variable is isolated.

Here is a simple guideline to solve equations:

• Keep the equation balanced by performing the same operations on both sides.
• Simplify progressively. Start by getting rid of constants on one side.
• Once you've isolated the term with the variable, solve for the variable by performing the reverse operation.

Imagine your equation as a balance scale. Whatever you do to one side, you must do to the other to maintain balance. This approach helps in reaching the correct solution systematically, as we saw in the original exercise.

Isolation of Variable

Isolation of the variable is a critical step in solving any equation. This means getting the variable by itself on one side of the equation. The main goal is to simplify the equation to the form where the variable equals a number.

Let's break down the steps on how to isolate a variable:

• To begin with, remove any constants from the side of the equation containing the variable. For example, in the equation \(7x - 5 = -54\), you would add 5 to both sides.
• Next, eliminate any coefficients attached to the variable by performing the inverse operation. In this case, divide both sides of the equation by the coefficient of \(x\), which is 7.

After performing these operations, you should reach a point where the variable, such as \(x\), stands alone on one side, making it easy to see its value in the equation.