Problem 111 - Algebra A Combined Function

Question

Decide whether the given number is a solution of the given equation. \(-3 x-5=-20 ; 5\)

Step-by-Step Solution

Answer

Yes, the number 5 is a solution to the equation.

Step 1: Substitute

Substitute the given number into the equation. Here, substitute 5 for \(x\) in the equation \(-3x - 5 = -20\). You will get \(-3(5) - 5 = -20\).

Step 2: Solve the Expression

Calculate the result of the expression \(-3(5) - 5\). First, \(-3(5) = -15\). Then, \(-15 - 5 = -20\).

Step 3: Compare with the Equation

Compare the result of the expression with the right side of the equation. You calculated \(-20\); since \(-20 = -20\), the equation holds true.

Key Concepts

Algebra

Algebra is a branch of mathematics that involves using symbols and letters to represent numbers and quantities in equations and expressions. It's a powerful tool for solving problems, especially when the exact values of some numbers are unknown. In algebra, we often use variables, like \(x, y, z\), to stand in for unknown numbers.

• Algebra allows us to create general formulas that can be applied to many situations, by manipulating these variables according to specific rules.
• Equations such as \(-3x - 5 = -20\) are central to algebra, where the goal is to find the value of the variable that makes the equation true.

Algebra helps in structuring problems as equations, making complex problems more manageable by breaking them down into simpler parts. It's the foundation for more advanced topics like calculus, and is essential in various fields such as science, engineering, and economics.

Substitution Method

The substitution method is a common technique used to find the value of an unknown variable in an equation. In this method, we replace the variable with a number or another expression. This helps in simplifying or solving the equation to see if the number is indeed a solution.

• To use substitution, you need a number you're checking as a potential solution for the equation.
• For instance, in the equation \(-3x - 5 = -20\), substituting \(5\) for \(x\) gives us \(-3(5) - 5\).

After substitution, we evaluate the expression to see if it equals the other side of the equation. This involves resolving any multiplication, division, addition, or subtraction in the expression. Substitution is particularly helpful in verifying whether a specific number is a solution to the given equation.

Linear Equations

Linear equations are a type of equation where each term is either a constant or the product of a constant and a single variable. These equations are called "linear" because they graph as straight lines.

• A basic example of a linear equation is \(-3x - 5 = -20\).
• The variables in linear equations like \(x\) do not have exponents other than 1, ensuring a straight-line graph.

Solving linear equations involves isolating the variable on one side of the equation to find its value. This can be done through operations like addition, subtraction, multiplication, or division, maintaining balance on both sides of the equation. Understanding linear equations is a key aspect of algebra, as they form the backbone of many real-world problem-solving scenarios.

Equation Verification

Equation verification is the process of checking whether a chosen value satisfies a given equation. This process confirms if we've accurately solved the equation or correctly proposed a solution.

• In our example, after substituting \(5\) into \(-3x - 5 = -20\), solving it gives \(-3(5) - 5 = -20\).
• When the calculated result \(-20\) matches with the given part of the equation \(-20\), verification is successful.

Equation verification ensures that our solution process is free from errors. It's crucial in mathematical problem-solving, ensuring confidence in results, especially in more complex contexts like physics or engineering where precise calculations are critical. By verifying equations, we confirm the logical steps of algebraic manipulation.