Problem 14 - Algebra 1

Question

Check whether the given number is a solution of the equation. $$\frac{1}{2} x-7=-4 ; 6$$

Step-by-Step Solution

Answer

Yes, the given number 6 is a solution of the equation because both sides yield -4.

Step 1: Substitute x Value

Substitute the value of \(x\) in the equation. In this instance \(x\) = 6. So, \(\frac{1}{2} * 6 - 7\) needs to be calculated

Step 2: Calculate the Value

Now, calculate it out: \(\frac{1}{2} * 6 - 7 = 3 - 7 = - 4\)

Step 3: Comparison of Result and Given Value

Finally, the result obtained should be compared against the given value on the right side of the equation. As we see, both equal -4

Key Concepts

Substitution Method

The substitution method is a fundamental technique in algebra used to find the value of one variable in a system of equations. It involves replacing a variable with its corresponding numerical value or another expression. This method is particularly useful when checking if a number is a solution to an equation, as seen in the given exercise.

For instance, if an equation looks like \(\frac{1}{2} x-7=-4\), and we wish to check if the number 6 is indeed a solution, we would 'substitute' the variable \(x\) with the number 6. The process of substitution makes equation solving seem like a simple evaluation of algebraic expressions. This method is integral to the foundation of algebra and is widely used not only in basic equation solving but also in more complex algebraic manipulations involving systems of equations.

It is important to follow the substitution accurately, making sure that all instances of the variable are replaced with the given number, as skipping this can lead to errors in the final result.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations. These expressions are the backbone of algebra, encapsulating the relationships between different quantities. For example, the expression \(\frac{1}{2} x - 7\) is an algebraic formation where the variable \(x\) represents an unknown quantity.

When working with algebraic expressions, it's essential to be comfortable with the basic operations: addition, subtraction, multiplication, and division, as well as understanding how to apply the order of operations, often remembered by the acronym PEMDAS ('Please Excuse My Dear Aunt Sally'), which stands for Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.

Understanding how to manipulate these expressions is crucial for solving equations and inequalities. It involves combining like terms, expanding expressions using the distributive property, and factoring, among other operations.

Equation Solving Steps

Equation solving is a process broken down into a series of steps to make finding the solution more manageable. The first step is often to simplify each side of the equation, which involves expanding expressions, combining like terms, or reducing fractions.

In the exercise provided, the steps include:

Substitute x Value

Here, we begin by substituting the provided number for \(x\).

Calculate the Value

Once substitution is done, the next step is to perform the calculation which simplifies the expression.

Comparison of Result and Given Value

Finally, the result is compared with the value on the other side of the equation to check for correctness.

It is important to approach each step methodically to avoid mistakes, and if at any step the equation is altered, ensuring balance is maintained by performing the same operation on both sides.