Problem 22 - Algebra for College Students

Question

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction. \(9 x+1=7 x-9\)

Step-by-Step Solution

Answer

The solution is \(x = -5\). The equation has been verified and is neither an identity nor a contradiction.

Step 1: Move all terms involving x to one side

Subtract 7x from both sides of the equation to get all the terms involving x on one side.\(9x + 1 - 7x = 7x - 9 - 7x\)This simplifies to:\(2x + 1 = -9\)

Step 2: Isolate the variable x

Subtract 1 from both sides to isolate the term with the variable x.\(2x + 1 - 1 = -9 - 1\)This simplifies to:\(2x = -10\)

Step 3: Solve for x

Divide both sides by 2 to solve for x.\(\frac{2x}{2} = \frac{-10}{2}\)This simplifies to:\(x = -5\)

Step 4: Check the solution

Substitute \(x = -5\) back into the original equation to verify the solution.Original equation: \(9x + 1 = 7x - 9\)Substitute \(x\) with \(-5\):\(9(-5) + 1 = 7(-5) - 9\)This simplifies to:\(-45 + 1 = -35 - 9\)Further simplifying:\(-44 = -44\)Since both sides are equal, the solution is verified.

Key Concepts

linear equations

Linear equations are mathematical statements that show two expressions are equal. They involve variables, often denoted as x or y, with a degree of one. The format of a linear equation is typically written in the form ax + b = c, where a, b, and c are constants. In the given exercise, we have the equation: 9x + 1 = 7x - 9. This is a linear equation because the variable x appears with an exponent of 1, making it a first-degree equation. Understanding linear equations is fundamental as they appear frequently in algebra and many practical applications.

algebraic manipulation

Algebraic manipulation is the process of rearranging and simplifying algebraic expressions to solve equations or inequalities. In our exercise, algebraic manipulation is used to isolate the variable x. We perform several operations:

• Subtracting 7x from both sides to get terms involving x on one side: 9x + 1 - 7x = 7x - 9 - 7x, resulting in 2x + 1 = -9
• Subtracting 1 from both sides to isolate the term with x: 2x + 1 - 1 = -9 - 1, leading to 2x = -10
• Dividing both sides by 2 to solve for x: 2x / 2 = -10 / 2, giving x = -5

These steps involve adding, subtracting, and dividing, which are basic techniques in algebraic manipulation. Mastering these techniques is crucial for solving more complex equations.

solving for x

Solving for x means finding the value of the variable x that satisfies the equation. In our example, we started with the equation 9x + 1 = 7x - 9. To solve for x, we:

• Moved all x terms to one side, resulting in 2x + 1 = -9
• Isolated the x term by subtracting 1, leading to 2x = -10
• Found the value of x by dividing both sides by 2, which gives x = -5

Each of these steps helps to progressively simplify the equation until we isolate x. Once isolated, the value of x can be determined. Solving for x is a fundamental skill in algebra that allows us to find unknown values in equations.

checking solutions in algebra

Checking solutions in algebra is a critical final step to ensure the correctness of our solution. After solving for x, we substitute x back into the original equation to verify if both sides are equal. In the exercise, we found x = -5 and substituted it back into the equation:

• Original equation: 9x + 1 = 7x - 9
• Substitute x = -5: 9(-5) + 1 = 7(-5) - 9
• Simplify: -45 + 1 = -35 - 9, leading to -44 = -44

Since both sides are equal, our solution x = -5 is verified as correct. This step ensures there are no errors in our calculations and that our solution is valid. It is an essential step in solving any algebraic equation.