Problem 49 - Algebra 2

Question

Graph each equation. $$ \frac{3}{5} y-\frac{x}{5}=-\frac{6}{5} $$

Step-by-Step Solution

Answer

Plot the y-intercept (0, -2), then another point (3, -1), and draw the line through these points.

Step 1: Solve for y

Isolate y on one side of the equation for easy graphing. Multiply each term by 5 to eliminate the fraction: \( 5(\frac{3}{5}y - \frac{x}{5}) = 5(-\frac{6}{5}) \), which simplifies to \( 3y - x = -6 \). Now solve this for y: \( y = \frac{x}{3} - 2 \).

Step 2: Identify the y-intercept

From the equation \( y = \frac{x}{3} - 2 \), the y-intercept occurs when \( x = 0 \), which gives \( y = -2 \). This is the point (0, -2) on the graph.

Step 3: Find another point using a different x-value

Choose a convenient value for x, preferably a multiple of 3 due to the fraction in the equation. Let's choose \( x = 3 \). Substituting into \( y = \frac{x}{3} - 2 \), we get \( y = 1 - 2 = -1 \), giving us the point (3, -1).

Step 4: Plot the points and draw the line

Plot the points (0, -2) and (3, -1) on a coordinate plane. Since two points determine a line, draw a straight line through these points. This is the graph of the given equation.

Key Concepts

Solving for Y

When graphing a linear equation, it's often useful to solve for the variable 'y' first. This process involves rearranging the equation so that 'y' is on one side all by itself. Doing so makes it simpler to understand how 'y' changes in relation to 'x'.
In the example \( \frac{3}{5}y - \frac{x}{5} = -\frac{6}{5} \), the first step is to clear the fractions by multiplying every term by 5, leading to \( 3y - x = -6 \). You can then rearrange the equation to isolate 'y', resulting in \( y = \frac{x}{3} - 2 \). This form of the equation, where 'y' is expressed in terms of 'x', is called the slope-intercept form, and it emphasizes the relationship between the variables.

Y-intercept

The y-intercept is where the graph of an equation crosses the y-axis on a coordinate plane. At this point, the value of 'x' is zero. Knowing the y-intercept is crucial when graphing, because it gives you a starting point.
For the equation \( y = \frac{x}{3} - 2 \), to find the y-intercept, we substitute zero for 'x', revealing that \( y = -2 \), which corresponds to the point (0, -2) on the graph. Understanding and identifying the y-intercept allows for quick and accurate placement of the first point when sketching the graph.

Coordinate Plane

The coordinate plane is a two-dimensional space formed by two number lines that meet at a right angle, creating an 'x-axis' and 'y-axis'. Points on this plane are determined by their position along these axes, with each point represented by an ordered pair (x, y).
To graph an equation on the coordinate plane, you begin by plotting known points, such as the y-intercept. For the equation we're working with, we first plot the point (0, -2), and then we choose another value for 'x' to find a second point, such as (3, -1) in this example. These points are essential for visualizing the linear relationship depicted by the equation.

Linear Equations

Linear equations are algebraic equations in which each term is either a constant or the product of a constant and a single variable. Linear equations form straight lines when graphed on a coordinate plane.
The standard form of a linear equation is \( Ax + By = C \), but the slope-intercept form \( y = mx + b \), where 'm' is the slope and 'b' is the y-intercept, can be more intuitive for graphing. By converting \( 3y - x = -6 \) to \( y = \frac{x}{3} - 2 \), we've made it easier to identify the slope ('rise over run') and y-intercept, two fundamental properties that define the line's graph.