Problem 12 - Algebra 1

Question

Write an equation in standard form of the line that passes through the given point and has the given slope. $$(2,5), m=3$$

Step-by-Step Solution

Answer

The equation of the line in standard form is \( 3x - y = 1 \)

Step 1: Use slope and point to write in slope-intercept form

We know the point (2,5) lies on the line, therefore when \( x = 2 \), \( y = 5 \). Substituting these into the slope intercept form \( y = mx + b \), where \( m = 3 \) is the given slope, we obtain \( 5 = 3 * 2 + b \). Solving this equation will give us the value of \( b \).

Step 2: Solve for the y-intercept b

Let's solve for \( b \) in the equation obtained in step 1, i.e., \[ 5 = 6 + b \]. Subtracting 6 from both sides of the equation gives \( b = -1 \). So, the equation of the line in slope-intercept form is \( y = 3x - 1 \).

Step 3: Convert to standard form

Standard form of equation is \( Ax + By = C \). From \( y = 3x - 1 \), moving terms to left side we get \( -3x + y = -1 \). However, for the standard form, 'A' needs to be positive, so we multiply by -1, to obtain: \( 3x - y = 1 \)

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is crucial when you're learning about the equations of lines. It is expressed as the equation \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) denotes the y-intercept—that is, the point where the line crosses the y-axis. This form is highly beneficial as it allows you to quickly sketch the line on a graph and understand its characteristics at a glance.

For instance, when you are given a point on a line, say \( (2,5) \) in our exercise, and the slope \( m = 3 \) you can directly substitute these values into the slope-intercept equation to determine the y-intercept \( b \). After substitution, you end with an equation that vividly depicts how a line ascends or descends across a graph as you move from left to right.

Solving for Y-Intercept

The y-intercept \( b \) plays a vital role in graphing linear equations and understanding the points of intersection with the y-axis. When you're given a specific point that the line passes through, alongside the slope of the line, you'll need to find \( b \) to fully define the line's equation in slope-intercept form.

To unearth the y-intercept, you insert the known point and slope into the slope-intercept form and solve for \( b \) by isolating it on one side of the equation. In the example you came across, the equation \( 5 = 3 \times 2 + b \) after substituting \( (2, 5) \) for \( x \) and \( y \) respectively and \( 3 \) for the slope \( m \) was solved by subtracting \( 6 \) from both sides, which left us with \( b = -1 \). This brief step exhibits how simple algebra can find this key coordinate.

Equation of a Line

The equation of a line can be expressed in various forms, including slope-intercept form, point-slope form, and standard form. Each form has its own applications and benefits, depending on the context. In many cases, you might need to convert between these different forms—like for graphing or meeting specific equation criteria.

Standard form is typically written as \( Ax + By = C \) and is useful, particularly when dealing with integer coefficients. In the scenario provided, after finding the slope-intercept form of the line, you can rearrange it into standard form by moving all terms involving variables to one side and keeping the constant on the other side. Remember to maintain the convention of the \( A \) coefficient being positive. From \( y = 3x - 1 \), by rearranging, we arrived at \( -3x + y = -1 \) and, to ensure \( A \) was positive, we multiplied through by \( -1 \) yielding the final standard form \( 3x - y = 1 \) as illustrated in the done exercise.