Problem 42 - A Graphical Approach to College Algebra

Question

Write equation in the form \(y=m x+b .\) (A suggested window for a comprehensive graph of the equation is given. \(2 y-5 x=0\) \([-10,10]\) by \([-10,10]\)

Step-by-Step Solution

Answer

The equation in slope-intercept form is \(y = \frac{5}{2}x\).

Step 1: Identify the Standard Form

The given equation is \(2y - 5x = 0\). This is in the standard form \(Ax + By = C\). Here, \(A = -5\), \(B = 2\), and \(C = 0\).

Step 2: Solve for y

We need to solve the equation for \(y\) to convert it to the slope-intercept form \(y = mx + b\). Start by isolating \(y\):\[2y = 5x.\]

Step 3: Simplify the Equation

Divide every term by 2 to solve for \(y\).\[y = \frac{5}{2}x.\] In this form, the slope \(m = \frac{5}{2}\) and the y-intercept \(b = 0\).

Step 4: Verify the Equation

Check whether the simplified equation \(y = \frac{5}{2}x\) is correct by substituting back into the original equation. Substitute \(y = 2.5x\) into \(2y - 5x = 0\) and see if both sides are equal, confirming our rearrangement.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most straightforward ways to write an equation of a line. Its general format is given as \( y = mx + b \), where:

• \( m \) represents the slope of the line, which indicates the steepness or tilt of the line. It shows how much \( y \) increases when \( x \) increases by 1 unit.
• \( b \) denotes the y-intercept, the point where the line crosses the y-axis.

Using this form, you can easily plot a line on a graph because you know where it begins (the y-intercept) and how it increases or decreases (the slope). In our exercise, the line was transformed into this form as \( y = \frac{5}{2}x \), where the slope \( m = \frac{5}{2} \) and the y-intercept \( b = 0 \).

Standard Form

Standard form is another way to present the equation of a line. The general structure here is \( Ax + By = C \), where:

• \( A \), \( B \), and \( C \) are integers (whole numbers), and \( A \) should ideally be positive.
• \( A \) and \( B \) are not both zero.

The standard form can often help in situations needing more direct calculations or when specific integer intercepts are needed. For instance, it’s beneficial when you want to determine the intercepts directly by setting \( x \) or \( y \) to zero. In our task, the equation \( 2y - 5x = 0 \) begins in standard form, identifying that \( A = -5 \), \( B = 2 \), and \( C = 0 \). From this form, we learned to convert it into the slope-intercept form by solving for \( y \).

Solving Equations

Solving equations is a crucial skill in understanding and working with linear equations. It involves manipulating an equation to find the value of a variable. In most linear equations, the aim is to solve for \( y \) or \( x \). To convert from standard to slope-intercept form, follow these steps:

• Isolate the variable \( y \) or \( x \) by adding, subtracting, multiplying, or dividing all terms to have \( y \) or \( x \) alone on one side of the equation.
• Perform operations carefully to maintain the equation's balance. Treat both sides of the equation equally with any operation.
• Ensure that the final equation represents \( y \) or \( x \) clearly, formatted as \( y = mx + b \) or \( x = my + b \), depending on which variable you solve for.

In our example, isolating \( y \) from \( 2y - 5x = 0 \) resulted in \( y = \frac{5}{2}x \), where \( y \) alone is on one side. This method of solving ensures clarity and makes plotting the equation on a graph straightforward.