Problem 119 - Beginning and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

Yes, 5 is a solution because substituting it satisfies the equation.

Step 1: Substitute the Value

To decide if 5 is a solution to the equation, substitute 5 for \(x\) in the equation:\[-3(5) - 5 = -20\]

Step 2: Simplify the Left Side

Calculate the expression by multiplying and then adding:\[-3 \times 5 = -15\]And then subtracting 5:\[-15 - 5 = -20\]

Step 3: Check Equality

Compare the simplified expression with the right side of the equation:\[-20 = -20\]Since both sides are equal, it shows that substituting \(x = 5\) satisfies the equation.

Key Concepts

Solving Equations

Equations are like balance scales. Each side needs to be equal for the scale to balance. When solving an equation, we want to find a number that, when substituted for the variable, makes both sides equal.
This process ensures that the equation is true at that particular value of the variable. For example, with the equation \(-3x-5=-20\), solving it means finding what number can be put in place of \(x\) so that the left side becomes equal to the right side, -20.
To solve, simply replace the variable with the number in question and check the calculation. If both sides of the equation are identical after substitution, then the number is indeed a solution.
Knowing how to solve equations is a fundamental skill in math. It helps to analyze, compare, and conclude if certain assumptions are correct. Solving equations is like solving puzzles!

Substitution

Substitution is a method used to determine if a specific number is a solution for an equation. It involves replacing the variable in the equation with a given number to see if the equation holds true.
In our specific problem with the equation \(-3x - 5 = -20\) and wondering if 5 is a solution, we perform substitution by replacing \(x\) with 5. After substitution, the equation transforms to \(-3(5) - 5 = -20\).
This step assesses whether our selected value satisfies the equation. If the equation balances (meaning the left side equals the right side), then 5 is confirmed as a solution. Substitution is vital as it helps verify proposed solutions and avoid guessing, ensuring the equation is solved correctly.

Simplification

Once you've substituted the variable in the equation, the next step is simplification. Simplification involves performing operations like addition, subtraction, multiplication, or division to condense complex expressions into simpler forms.
In the equation \(-3(5) - 5 = -20\), simplification is crucial. First, multiply and simplify \(-3 \times 5 = -15\). Then, proceed by subtracting 5 from -15 to get -20.
Simplification lets you clearly see if both sides of the equation match. It's like cleaning up a messy room – everything becomes clearer as things are put in place. Simplification is not just a matter of practice; it also builds a strong foundational understanding of how math operations interact with each other.