Problem 13 - Algebra 1

Question

Check whether the given number is a solution of the equation. $$7 x-15=-1 ;-2$$

Step-by-Step Solution

Answer

-2 is not a solution to the given equation.

Step 1: Substitute the given value into the equation

First of all, substitute -2 for x in the equation \(7x - 15 = -1\). This results in: \(7(-2) - 15 = -1\)

Step 2: Simplify the left side of the equation

Multiply 7 by -2 to get -14 and subtract 15 from -14 to get -29. So the left side of the equation becomes -29: \(-29 = -1\)

Step 3: Check the equation

Now check whether the equation holds true. It's seen that -29 is not equal to -1, so the equation does not hold true. Therefore, -2 is not a solution to the equation.

Key Concepts

Substitution Method

The substitution method is a pivotal technique used in algebra to solve systems of equations. However, it's also instrumental when we aim to check if a given number is a solution to an equation. By replacing the variable with the given number, we achieve a concrete numerical expression which can be evaluated.

For example, if we have the equation \(7x - 15 = -1\), and we want to test whether -2 is a solution, we substitute -2 for \(x\). This substitution results in a new equation solely containing numbers, eliminating the variable entirely. The key advantage here is that we transform the abstract equation into an arithmetic problem that can be solved using basic operations—addition, subtraction, multiplication, and division. Always remember to perform the substitution accurately to avoid calculation mistakes that can lead to incorrect conclusions about the solution's validity.

Equation Simplification

Simplification of an equation is much like tidying up a cluttered room – it's about making the problem as straightforward as possible to solve. After substituting the value into the equation, as with our earlier example where we input -2 into \(7x - 15 = -1\), simplification comes next. You'll start with arithmetic – multiply 7 by -2 to end up with -14, then subtract 15 from -14 to get -29.

This step is crucial as it allows you to see, in plain terms, if the left side of the equation matches the right side. In this case, simplifying left us with \(-29 = -1\), which is clearly not true. It's essential to carry out each arithmetic operation correctly, following the order of operations, also known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), to ensure the simplification is done correctly.

Verifying Solutions

Verifying solutions serves as the 'proof' in mathematics - it's how we ascertain the accuracy of our work. After substituting and simplifying the equation, we compare our simplified expression to the other side of the equation. If these two are equivalent, our initial guess (the value we substituted) is indeed a solution; if not, it's back to the drawing board. In our example problem, after simplification, we established that \(-29\) does not equal \(-1\).

Verification is the final checkpoint in ensuring that we haven't made any mistakes along the way. It's not enough for an equation to appear correct; it must be validated through this process of verification. To improve your skills, practice by checking the solutions with different numbers and equations. Verification builds your confidence in your mathematical abilities and helps solidify your understanding of algebraic concepts.