Problem 48 - Algebra and Trigonometry

Question

Find the slope and y-intercept of the line, and draw its graph. $$ 2 x-5 y=0 $$

Step-by-Step Solution

Answer

Slope is \( \frac{2}{5} \), y-intercept is 0, and line passes through (0,0) and (5,2).

Step 1: Understand the Standard Form Equation

The given equation is in the standard form, which is generally written as \( Ax + By = C \). Here, \( A = 2 \), \( B = -5 \), and \( C = 0 \).

Step 2: Convert to Slope-Intercept Form

The slope-intercept form of a line is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. We need to rearrange the equation \( 2x - 5y = 0 \) to match this form.

Step 3: Solve for y

Subtract \( 2x \) from both sides to isolate the \( y \)-term: \(-5y = -2x\).

Step 4: Divide by the Coefficient of y

Divide all terms by \(-5\) to solve for \( y \): \( y = \frac{2}{5}x\). The equation is now in slope-intercept form.

Step 5: Identify the Slope and y-intercept

In the equation \( y = \frac{2}{5}x \), the slope \( m = \frac{2}{5} \) and the y-intercept \( b = 0 \).

Step 6: Draw the Graph

To graph the line, start at the y-intercept \((0, 0)\) on the coordinate plane. Use the slope \( \frac{2}{5} \) to find another point: from \( (0, 0) \), move up 2 units and right 5 units to reach the point \((5, 2)\). Draw the line through these points.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is fundamental to understanding how lines behave on a graph. This form is often written as \( y = mx + b \). Here, \( m \) represents the slope of the line, which tells us how steep the line is, and \( b \) indicates the y-intercept, which is the point where the line crosses the y-axis. Converting other forms of linear equations to this form can make it easier to quickly identify these characteristics.

Why is it useful? Not only does it let you graph a line quickly by plotting the y-intercept and using the slope, but it also allows you to see the relationship between variables at a glance. For example, a positive slope means the line rises as you move from left to right, while a negative slope means it falls.

In our exercise, we converted the equation \( 2x - 5y = 0 \) to \( y = \frac{2}{5}x \). This conversion shows a slope, \( m = \frac{2}{5} \), and a y-intercept, \( b = 0 \). This tells us that for every 5 units we move to the right, the line will rise by 2 units.

Graphing Linear Equations

Graphing linear equations involves plotting points on a coordinate plane and drawing a straight line through those points. Once the equation is in slope-intercept form, it's straightforward.

Here's a simple way to graph using \( y = mx + b \):

• Start by finding the y-intercept \( b \). Place your first point on the y-axis at \( y = b \).
• Use the slope \( m \) to find another point. The slope \( \frac{rise}{run} \) tells you how many units to go up or down (rise) and how many to move right (run).
• For our equation \( y = \frac{2}{5}x \), start at \( (0,0) \) since \( b = 0 \).
• From there, move up 2 units and right 5 units to plot \( (5, 2) \).

Draw a line through these points, and extend it across the graph. Each point on the line represents a solution to the equation.

Checking your work is easy. Pick any point on the line, and substitute the x and y-values into the original equation to see if they satisfy it.

Standard Form of a Linear Equation

The standard form of a linear equation is \( Ax + By = C \). This form is quite versatile and has particular benefits in various scenarios:

• It's great for finding intersections and dealing with vertical or horizontal lines.
• You can easily see the equation's balance and intuitively adjust the values of \( A \), \( B \), or \( C \).

In our exercise, the given equation was \( 2x - 5y = 0 \), where \( A = 2 \), \( B = -5 \), and \( C = 0 \). This form is useful for quickly recognizing symmetries and balancing, though it's not immediately helpful for graphing.

To convert to slope-intercept form, you typically solve for \( y \). This might involve rearranging the equation to isolate the y-term and then dividing through by the coefficient of \( y \). In our example, we did precisely this to make graphing straightforward. Understanding how to manipulate between different forms of linear information makes mathematics more flexible and powerful.