Problem 65 - Algebra A Combined Function

Question

Solve each equation for y. See Section 2.5. $$ 10 x=-5 y $$

Step-by-Step Solution

Answer

The solution is \(y = -2x\).

Step 1: Isolate the Variable

First, we want to isolate the variable \(y\) on one side of the equation. The equation is given as\[ 10x = -5y \]. To do this, divide both sides of the equation by \(-5\):\[ \frac{10x}{-5} = y \].

Step 2: Simplify the Expression

Now, simplify the expression on the left side. The division \(\frac{10x}{-5}\) can be simplified by dividing 10 by -5:\[ y = -2x \].

Key Concepts

Isolating the Variable

To isolate a variable in an algebraic equation means to get the variable by itself on one side of the equation. This is crucial for solving equations because it allows you to find the value of the variable you're interested in.
In the exercise given, the goal is to isolate the variable \( y \) from the equation \( 10x = -5y \). To do that, you want \( y \) alone on one side. You can achieve this by performing operations that will cancel out the other terms along with \( y \).

• Identify the term containing \( y \), which is \(-5y\).
• Perform the opposite operation on both sides of the equation to cancel out \(-5\). Since it’s multiplied by \( y \), you divide both sides by \(-5\).

Once you've divided by \(-5\), \( y \) stands alone, completing your isolation step with \( y = \frac{10x}{-5} \). Isolating variables is like peeling away layers until all you're left with is the variable on one side.

Simplifying Expressions

Simplifying expressions involves performing basic arithmetic operations to make the equation as neat and straightforward as possible. After isolating \( y \), you end up with \( y = \frac{10x}{-5} \). Here, the expression is not in its simplest form yet.To simplify, you need to divide the numerator by the denominator, which involves:

• Dividing the coefficent of \( x \), which is 10, by \(-5\).

When you perform the operation, \( 10 \div -5 \), you get \(-2\). This means the simplified form of \( y \) is \( y = -2x \).Simplifying expressions helps in making them easier to understand and work with. It turns more complex computations into simpler arithmetic.

Linear Equations

Linear equations are equations that graph as straight lines. They often have two variables, and both are raised to the first power. The solution for these equations is a set of points that form a line.In the given equation \( y = -2x \), this is already in the form of a linear equation. Linear equations are usually expressed in the form of \( y = mx + b \), where:

• \( m \) is the slope of the line.
• \( b \) is the y-intercept.

In this exercise, \( y = -2x \), and \( m = -2 \) while \( b = 0 \). This means the line crosses the origin, and for every increase of 1 in \( x \), \( y \) decreases by 2 units. Understanding linear equations helps in visualizing how changes in \( x \) affect \( y \), and it is fundamental in analyzing relationships within data.