Problem 19 - College Algebra and Calculus: An Applied Approach

Question

Solve the equation and check your solution. (Some equations have no solution.) $$ 7-2 x=15 $$

Step-by-Step Solution

Answer

The solution to the equation \(7 - 2x = 15\) is \(x = -4\).

Step 1: Isolate the term with the variable

The equation we are provided with is \(7 - 2x = 15\). The first step involves isolating the expression containing the variable \(x\). To do this, subtract 7 from both sides of the equation. This gives us \(-2x = 15 - 7\) or \(-2x = 8\).

Step 2: Solve for the variable x

We now have \(-2x = 8\). To solve for \(x\), we need to get rid of the -2 coefficient in front of \(x\). We do this by dividing both sides of the equation by -2. Doing this gives us \(x = 8 / -2\) or \(x = -4\).

Step 3: Check the solution

To confirm if \(x = -4\) is the correct solution, we substitute it back into the original equation. Thus, the left side of the equation becomes \(7 - 2(-4) = 7 + 8 = 15\). Since this is equal to the right side of the original equation, the solution is correct.

Key Concepts

Isolating the Variable

Isolating the variable is a crucial step when solving linear equations. It involves manipulating the equation so that the variable stands alone on one side. This makes it much easier to solve. Let's consider the example equation given:
\[7 - 2x = 15\] To isolate the expression containing \(x\), we want to move everything except for the term with \(x\) to the other side of the equation. In this case, it's the number 7:

• Subtract 7 from both sides: \(-2x = 15 - 7\)
• Simplify the equation: \(-2x = 8\)

These actions eliminate the constant term from the side containing the variable, allowing us to focus solely on \(x\). Remember, whatever you do to one side of the equation, do to the other. This is fundamental in maintaining balance and ensuring the equation's integrity.

Checking Solutions

Verifying your solution is an important concluding step. After finding a potential solution for \(x\), it's vital to check if it truly satisfies the original equation. This ensures the accuracy of your answer, and alerts you to any possible mistakes you might have made while calculating.
Here's how you would check your solution \(x = -4\):

• Substitute \(x = -4\) back into the original equation: \(7 - 2(-4)\)
• Simplify the calculation: \(7 + 8 = 15\) because \(-2 \times -4 = 8\)
• Check if both sides of the equation match: \(15 = 15\)

Since the left side equals the right side, the solution is verified as correct. If the two sides do not equal, it's a sign there might have been an error in your calculations.

Linear Algebra Concepts

In solving linear equations, you're engaging with fundamental principles of linear algebra. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable.
Let's break down some essential concepts:

• **Coefficients:** These are numbers in front of variables. For example, in \(-2x\), \(-2\) is the coefficient.
• **Operations:** Addition, subtraction, multiplication, and division can be applied, but must be done equally to both sides to maintain the equation's balance.
• **Solution:** Solving a linear equation often involves figuring out what the variable must be in order to make the equation true. In our example, the solution is \(x = -4\).

These concepts are the building blocks of linear algebra and are used widely in more complex math problems. Mastering them allows you to understand and solve a wide array of mathematical equations and problems efficiently.