Problem 32 - Algebra 1

Question

Solve the equation. $$-10 x=-9$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 0.9\).

Step 1: Identify the equation

We have the equation \(-10x = -9\). Our goal is to solve for x.

Step 2: Isolate the Variable

To isolate x, we can divide both sides of the equation by -10. This gives us \(x = \frac{-9}{-10}\).

Step 3: Simplify the result

Simplify the right side. This results to \(x = 0.9\).

Key Concepts

Isolating the Variable

In algebra, to solve for a variable means to find the value for the variable that makes the equation true. This process often involves 'isolating the variable'—a critical step that sets the variable on one side of the equation and the constants on the other. One common method to achieve isolation is by performing the same operation on both sides of the equation, ensuring that the equation remains balanced.

Take, for example, the equation we have here: \( -10x = -9 \). To isolate \(x\), we want to get \(x\) by itself on one side. In this case, we can divide both sides by -10 because \(x\) is being multiplied by -10. This operation will cancel out the -10 on the left, leaving \(x\) alone. Therefore, \( x = \frac{-9}{-10} \).

Isolating variables can include different operations such as addition, subtraction, multiplication, or division, depending on what the equation requires. It's also important to perform the operations in the correct order, remembering the rules of algebraic operations, and considering the goal of isolation to make the variable the subject of the formula.

Simplifying Expressions

Once we have isolated the variable, it is often necessary to 'simplify the expression' to find the variable's value in its simplest form. Simplification can involve various procedures such as reducing fractions, combining like terms, or applying mathematical properties like the distributive property.

For instance, after isolating \(x\) in our given equation and achieving the fraction \( \frac{-9}{-10} \), we simplify this fraction to its decimal form to make it more interpretable. Simplification doesn't change the value of the expression; it simply makes it easier to understand or use in further calculations. Here, our simplified result is \(x = 0.9\), which is the solution to the equation in decimal form.

It's crucial to know when an expression is fully simplified; for a fraction, this means it cannot be reduced any further. For decimals, it might mean rounding to a certain number of decimal places as directed, or it might mean recognizing a repeating decimal.

Equation Solving Steps

Solving any linear equation can be broken down into a series of steps that, if followed correctly, will lead to the solution. The steps begin with simplifying both sides of the equation if necessary, by combining like terms and reducing expressions to their simplest form. Then, isolate the variable by performing operations that move all instances of the variable to one side of the equation.

The general steps for solving linear equations include:

• Identify and write down the problem.
• Simplify expressions on both sides of the equation.
• Use inverse operations to isolate the variable.
• Check the solution by substituting it back into the original equation.

Our example equation, \( -10x = -9 \), is already in a simplified form, so we started with isolating \(x\). After dividing by -10 and simplifying, we identified \(x = 0.9\) as the solution. To confirm the solution is correct, we substitute \(x = 0.9\) back into the original equation to see if the left and right sides are equal—a fundamental step in validating our answer. Always remember, checking the solution is as necessary as finding the solution itself, as it ensures accuracy in the problem-solving process.