Problem 29 - Algebra 1: Concepts and Skills

Question

SOLVING EQUATIONS Use division to solve the equation. $$ -10 x=-9 $$

Step-by-Step Solution

Answer

The solution to the equation \(-10x = -9\) is \(x = 0.9\)

Step 1: Understand the equation

The given equation is \(-10x = -9\). The objective here is to find the value of the variable \(x\) that makes the equation true.

Step 2: Apply division to both sides

The variable \(x\) is currently multiplied by -10. To isolate \(x\), divide both sides of the equation by -10.

Step 3: Simplify and solve for \(x\)

When you divide both sides by -10, the equation simplifies to \(x = \frac{-9}{-10} = 0.9\).

Key Concepts

Isolate the Variable

When solving algebraic equations, isolating the variable is a fundamental step. It means to 'get the variable by itself' on one side of the equation. In the given exercise, the equation is \( -10x = -9 \). The variable \( x \) is the unknown that we're trying to solve for, but it's currently tied up with a -10.

To isolate \( x \) we perform operations that will 'undo' the multiplication by -10. The inverse operation of multiplication is division, so we'll need to divide both sides of the equation by -10. By doing this, we're essentially balancing the equation, since any operation done to one side must also be done to the other to maintain equality. The act of isolating the variable lays down the pathway to easily find the solution without changing the meaning of the equation.

Division in Algebra

Division in algebra functions just as it does in basic arithmetic: it's used to distribute a number evenly into parts or to reverse multiplication. However, when dividing algebraic expressions, it's crucial to apply the division to the entire expression on both sides of the equation. This step is known as the division property of equality.

In the exercise's step 2, there's a need to divide by -10 to both sides of \( -10x = -9 \). It is essential to highlight that signs are important in algebra. Since both the coefficient of \( x \) and the constant on the other side are negative, the division will result in a positive number because dividing two negative numbers results in a positive. This rule is a vital concept in solving equations that involve negative numbers.

Simplifying Equations

Simplifying equations is an art in the realm of algebra. It means to reduce an equation to its simplest form, making it easier to solve. This involves combining like terms, reducing fractions, and carrying out arithmetic operations. In the final step of solving the given exercise, simplifying involves executing the division which has been set up in the previous steps.

Thus, after dividing both sides by -10, the equation becomes \( x = \frac{-9}{-10} \), which simplifies to \( x = 0.9 \). Here, noticing that a negative divided by a negative gives a positive is crucial to reaching the correct answer. The simplified form reveals the solution to the problem directly, showing that \( x \) equals 0.9. Simplification is essentially the step that paints the clear picture and presents the solution in its least complex form.