Problem 13 - Elementary Algebra

Question

In the following problems, solve each of the conditional equations. $$ 4 x=56 $$

Step-by-Step Solution

Answer

Answer: The solution of the equation 4x = 56 is x = 14.

Step 1: Write down the given equation

The given equation is: $$ 4x = 56 $$

Step 2: Isolate the variable x

To isolate x, we need to divide both sides of the equation by the coefficient of x, which is 4: $$ \frac{4x}{4} = \frac{56}{4} $$

Step 3: Simplify the equation

Now, simplify the equation by performing the division operation on both sides: $$ x = 14 $$ So, the solution for the given conditional equation is x = 14.

Key Concepts

Algebraic Equations

An algebraic equation is a statement of equality between two expressions that involve variables and constants. These equations play a crucial role in mathematics and various other fields, representing relationships between quantities. Essentially, they are puzzles we solve to find the unknown values.

For example, in the equation 4x = 56, the expressions on both sides of the equals sign represent the same quantity, and the goal is to find the value of x that makes this statement true. Algebraic equations can range from simple, linear expressions like the given example to more complex ones involving higher degree polynomials, exponents, and more.

The process of solving these equations includes techniques such as simplification, isolation of the variable, and using operations that maintain the equality such as adding, subtracting, multiplying, and dividing both sides of the equation by the same number or expression.

Isolate Variable

To isolate a variable means to manipulate an equation in such a way that the variable of interest stands alone on one side of the equation, with a coefficient of 1. This is often a key step in solving algebraic equations as it gives us the value of the variable directly.

In the case of our example equation, 4x = 56, isolating the variable x involves dividing both sides of the equation by 4, which is the coefficient of the variable. The operation can be seen as peeling away the layers that conceal the value of x, until only x remains by itself on one side of the equation, providing a clear answer. It's like finding the 'x' that holds the balance of the scales perfectly even.

Simplification

Simplification is a method of reducing an algebraic equation to its simplest form in order to make the solution clearer and easier to understand. In simplification, we perform all the possible arithmetic operations to condense the equation.

It involves combining like terms, reducing fractions, and clearing parentheses among other actions. With our given equation, once we divide both sides by the coefficient 4, we simplify it to x = 14. This simpler equation readily shows the solution without extraneous details, making it easy to verify that this value of x satisfies the initial equation.

Coefficient

A coefficient in an algebraic equation is the numerical factor of a term that contains a variable. In simpler terms, it's the number directly in front of a variable, representing how many times that variable is being multiplied.

For instance, in 4x = 56, the number 4 is the coefficient of x. Understanding coefficients is crucial when solving equations because they determine how the variable is scaled or weighted. When isolating the variable, as we did to solve for x, we manipulated the coefficient to set the variable free, so to speak, arriving at its value. Recognizing and working with coefficients correctly are foundational skills in algebra that help students unlock the solutions to countless algebraic puzzles.