Question

Find the slope and y-intercept of each line. Graph the line. $$ \frac{1}{3} x+y=2 $$

Short answer

Slope: \( -\frac{1}{3} \), y-intercept: 2

Step-by-step solution

Convert Equation to Slope-Intercept Form

The slope-intercept form of a line is given by the equation: \[ y = mx + b \]where \( m \) is the slope and \( b \) is the y-intercept. Start by converting the given equation to this form.

Isolate y on One Side

To isolate \( y \), subtract \( \frac{1}{3} x \) from both sides of the equation: \[ y = -\frac{1}{3} x + 2 \].

Identify the Slope and Y-Intercept

From the equation \( y = -\frac{1}{3} x + 2 \), identify: Slope (\( m \)): \( -\frac{1}{3} \)Y-Intercept (\( b \)): \( 2 \)

Graph the Line

Plot the y-intercept (0,2) on the graph. Use the slope \( -\frac{1}{3} \), which means for each step right (positive x-direction), go down by \( \frac{1}{3} \) units. Plot another point using this slope and draw the line through both points.

Key concepts

Slope-Intercept Form

The slope-intercept form is a way to write linear equations. It makes it easy to find the slope and y-intercept. This form looks like: \text{y} = \text{mx} + \text{b}. Here, \text{m} is the slope and \text{b} is the y-intercept. This form is very useful because you can quickly see the slope and where the line crosses the y-axis. For example, consider the equation \(\text{y} = -\frac{1}{3} \text{x} + 2\). Here, the slope (\text{m}) is \(- \frac{1}{3}\) and the y-intercept (\text{b}) is \(2\). By just looking at this form, you immediately know these important details about the line.

Graphing Linear Equations

Graphing a linear equation lets you visually see the line which represents the solutions to the equation. To graph an equation in slope-intercept form (\text{y} = \text{mx} + \text{b}), follow these steps:

• Step 1: Locate the y-intercept (\text{b}) on the graph. This is where the line crosses the y-axis.
• Step 2: Use the slope (\text{m}) to find another point on the line. Remember, slope is rise over run. If \text{m} = \(-\frac{1}{3}\), it means you go down \(1\) unit and right \(3\) units from the y-intercept to plot the next point.
• Step 3: Connect these points with a straight line.

Let's use the equation \(\text{y} = -\frac{1}{3} \text{x} + 2\). First, plot the point \((0, 2)\) at the y-intercept. Next, use the slope to move down \(1\) unit and right \(3\) units to find another point, such as \((3, 1)\). Draw a line through these points to graph the equation.

Algebraic Manipulation

Algebraic manipulation is essential to solving and rearranging equations. To convert an equation into slope-intercept form, you often need to isolate \(\text{y}\) on one side. Start by moving terms involving \(\text{x}\) to the other side of the equation. Let's manipulate the equation \(\frac{1}{3}\text{x} + \text{y} = 2\). Subtract \(\frac{1}{3}\text{x}\) from both sides: \text{{y}} = -\frac{1}{3}\text{x} + 2. Now it's in slope-intercept form. Knowing how to rearrange equations like this helps to simplify problems and identify key values like slope and y-intercept.

Linear Equations

Linear equations represent straight lines on a graph. They generally have variables with exponents of 1. The basic form of a linear equation is \(\text{Ax} + \text{By} = \text{C}\), where \(\text{A}\), \(\text{B}\), and \(\text{C}\) are constants. These equations are useful in many areas of mathematics and real-world problems. For example, take the equation \(\frac{1}{3}\text{x} + \text{y} = 2\). To better understand and graph this equation, convert it to slope-intercept form: \(\text{y} = -\frac{1}{3}\text{x} + 2\). This transformation shows the slope and y-intercept, making it simpler to work with and visualize on a graph.