Problem 30 - Algebra 1

Question

Write an equation of the line that has the given \(x\) -intercept and slope. $$x \text { -intercept }=2, m=-\frac{2}{3}$$

Step-by-Step Solution

Answer

The equation of the line with \(x\) -intercept 2 and slope \(-2/3\) is \(y = -\frac{2}{3}x + \frac{4}{3}\).

Step 1: Identify the given point

The x-intercept is given to be \(2\). This means that the line intersects the x-axis at \(2\), so the point is given as \((2,0)\).

Step 2: Identify the slope

The slope is given as \(-\frac{2}{3}\). Slope refers to the 'steepness' of a line.

Step 3: Formulate the Equation

We will use the point-slope form of the line equation which is \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a point on the line and \(m\) is the slope. Substitution of the point \((2,0)\) and the slope \(-\frac{2}{3}\) into the equation gives us: \(y - 0 = -\frac{2}{3}(x - 2)\).

Step 4: Simplify the Equation

Simplifying the equation gives \(y = -\frac{2}{3}x + \frac{4}{3}\)

Key Concepts

X-Intercept

The x-intercept of a line is the point where the line crosses the x-axis on a coordinate plane. In essence, it is where the value of y is zero. When we are given the x-intercept, we are provided with a crucial piece of information that helps us to write the equation of the line. For example, if the x-intercept is 2, we know that one of the points on the line is (2, 0). This fixed point, along with the slope, enables us to construct the entire line.

Identifying the x-intercept is often the first step when tasked with writing the equation of a line. It is important to remember that the coordinates of the x-intercept will always have the form \( (x, 0) \), underscoring the idea that the y-value will invariably be zero at this point.

Slope

The slope of a line, often denoted by the letter 'm', represents how steep the line is. It is a measure of the rate at which the line rises or falls as we move along the x-axis. A positive slope means the line is inclining upwards, while a negative slope indicates that it's declining. The higher the absolute value of the slope, the steeper the line.

The slope is calculated by the difference in y-coordinates divided by the difference in x-coordinates between two points on the line, known as the rise over run. In our case, the given slope is \( -\frac{2}{3} \), indicating a line that falls as it moves from left to right.

Point-Slope Form

The point-slope form is a straightforward method for writing the equation of a line when you know a single point on the line and its slope. The formula for this form is \( y - y_1 = m(x - x_1) \), with \( (x_1, y_1) \) being the coordinates of the known point and 'm' being the slope.

To use this form effectively, plug in the slope for 'm' and the coordinates of the known point for \( x_1 \) and \( y_1 \) respectively. From there, simplify the equation to find the y-intercept, which is critical for graphing the line or converting the equation into other forms such as the slope-intercept form.

Linear Equations

A linear equation is an algebraic equation that represents a straight line when graphed on a coordinate plane. These equations can take several forms, such as slope-intercept form (\(y = mx + b\)), point-slope form, and standard form (\(Ax + By = C\)). The most commonly used is the slope-intercept form because it clearly shows the slope and the y-intercept of the line.

Understanding how to manipulate these forms is vital in algebra, as it allows you to graph the line, predict points along the line, and solve for various x and y values. Moreover, linear equations underpin many real-world relationships and, as such, are foundational in both mathematics and applied sciences.