Problem 58 - Algebra 1

Question

Write an equation of the line that passes through the point and has the given slope. Use slope-intercept form. $$ (-1,5), m=\frac{2}{3} $$

Step-by-Step Solution

Answer

The equation of the line is \(y=\frac{2}{3}x + \frac{17}{3}\)

Step 1: Use the given point and slope

The equation of a line using slope-intercept form is \(y = mx + b\). We substitute the given point \((-1,5)\) into the equation and the given slope \(\frac{2}{3}\), which would give us \(5 = \frac{2}{3}*(-1) + b\).

Step 2: Calculate the y-intercept

Then, we solve for \(b\) to determine the y-intercept. By simplifying the equation, we get \(b = 5 + \frac{2}{3}\) which equates to \(b=\frac{17}{3}\).

Step 3: Construct the final equation

Finally, we put the calculated \(b\) and the given \(m\) back into the slope-intercept form to get the equation in the format \(y=mx+b\), which results into \(y = \frac{2}{3}x + \frac{17}{3}\).

Key Concepts

Equation of a Line

The equation of a line is a mathematical way to represent a straight line on a graph. In algebra, one of the most popular forms of displaying this equation is the slope-intercept form. This form is helpful because it clearly shows two essential characteristics of a line: its slope and its y-intercept. The standard form for the equation of a line in slope-intercept form is \(y = mx + b\), where:\

• \( y \) is the output or dependent variable.
• \( x \) is the input or independent variable.
• \( m \) is the slope of the line.
• \( b \) is the y-intercept where the line crosses the y-axis.

Understanding how each part contributes to the overall equation is critical for graphing and interpreting linear functions. When you input different values of \( x \), this equation allows you to calculate their corresponding \( y \) values, which ultimately helps in plotting the linear path on a graph.

y-intercept

The y-intercept of a line is where the line crosses the y-axis on a graph. This point is crucial because it provides a fixed point from which the line begins its journey across the grid. In the equation of a line in slope-intercept form \(y = mx + b\), the term \(b\) represents the y-intercept.
If you have a graph, you can spot the y-intercept right where the line meets the vertical y-axis. This makes sense because the x-coordinate at this point would be zero, as the line hasn't yet moved horizontally across the axis.

• For example, if the y-intercept \(b\) is \( \frac{17}{3} \), our line meets the y-axis at \( y = \frac{17}{3} \).

Calculating the y-intercept requires substituting a known point and the slope into the linear equation and solving for \(b\). Understanding and identifying the y-intercept is essential for drawing accurate graphs and for understanding how a linear change starts in the coordinate plane.

Slope

The slope of a line is a measure of its steepness and direction, indicating how much \(y\) changes for a change in \(x\). Represented by \(m\) in the slope-intercept form \(y = mx + b\), the slope is vital in determining the line's tilt. It can be calculated as the rise over the run, which is \( \frac{\Delta y}{\Delta x} \), the change in y over the change in x.

• A positive slope means the line rises from left to right, while a negative slope means it falls.
• A zero slope signifies a horizontal line, and an undefined slope shows a vertical line.

For example, in our line equation, the slope \( \frac{2}{3} \) suggests for every three units moved right along the x-axis, the line ascends two units up the y-axis. By understanding slopes, you can predict the direction of a line and how quickly it rises or falls across a graph. This comprehension is essential for creating equations and graphs that depict real-world linear relationships.