Problem 13 - Algebra 1: Concepts and Skills

Question

Rewrite the equation in slope-intercept form. $$10 x-5 y=50$$

Step-by-Step Solution

Answer

The equation in slope-intercept form is \(y = 2x - 10\).

Step 1: Isolate terms involving y

First, subtract \(10x\) from both sides of the equation to isolate terms involving \(y\). This results in the equation \(-5y = -10x + 50\).

Step 2: Divide by the coefficient of y

Divide all terms of the equation by -5 to solve for \(y\). This gives the equation \(y = 2x - 10\).

Step 3: Check the Result

Check the result to make sure it's in slope-intercept form \(y = mx + b\). The equation \(y = 2x - 10\) has a slope \(m = 2\) and y-intercept \(b = -10\), which is exactly what we need.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is essential when dealing with linear equations. This special format expresses a line equation as \( y = mx + b \), where \( m \) represents the slope and \( b \) denotes the y-intercept of the line. The slope measures the steepness and direction of the line, and the y-intercept is the point where the line crosses the y-axis.

In practice, rewriting an equation into this form provides a quick way to graph the line and understand its characteristics. When you come across an equation like \( 10x - 5y = 50 \), transforming it into slope-intercept form makes it easier to visualize and work with. It's like translating a complex sentence into a simpler one that still conveys the same meaning but is much easier to grasp.

Isolating Variables

Isolating variables is a fundamental technique in algebra. It involves rearranging an equation to get one variable by itself on one side of the equation. The process is like solving a puzzle where you shift pieces—numbers and variables—around until the picture becomes clear.

Techniques for Isolation

To isolate a variable, you can add, subtract, multiply, or divide both sides of the equation by the same number. For example, in the equation \( 10x - 5y = 50 \), you would start by subtracting \( 10x \) from both sides, leading to \( -5y = -10x + 50 \). The goal is to untangle the variable you're solving for ( this case, \( y \)) from the other terms. After, dividing every term by the coefficient of y further isolates y, giving us \( y = 2x - 10 \). Each step moves you closer to the solution while maintaining the balance of the equation.

Solving Linear Equations

Solving linear equations is like finding the key to a lock. The equation represents a balance, and your job is to maintain that balance while finding the value of the unknown variable. Linear equations are the simplest type of equation and often resemble a straight line when graphed.

Keeping Equations Balanced

When you operate on a linear equation, whatever you do to one side, you must also do to the other—this keeps the equation balanced. In the given example, dividing both sides by -5 ensures that the balance is maintained, resulting in a solution for y. The end goal is to arrive at an equation where y is expressed as a function of x, which is the most straightforward relationship to understand and graph.

Remember to check your solution to ensure it fits the slope-intercept form. By mastering these techniques, you'll unlock the power of algebra and make sense of lines, slopes, and intercepts in no time.