Problem 14 - Prealgebra

Question

Complete the given ordered pairs, and use the results to graph the equation. (GRAPH CANT COPY) $$y=\frac{1}{3} x+1 \quad(-3,),(0,),(3,)$$

Step-by-Step Solution

Answer

The ordered pairs are (-3, 0), (0, 1), and (3, 2). Plot these on a graph and draw a line through them.

Step 1: Determine y for x = -3

First, we need to find the y-coordinate when the x-coordinate is -3 using the equation \( y = \frac{1}{3}x + 1 \). Substitute \( x = -3 \) into the equation: \[y = \frac{1}{3}(-3) + 1\]This simplifies to: \[y = -1 + 1 = 0\].Therefore, the ordered pair is \((-3, 0)\).

Step 2: Determine y for x = 0

Now, calculate the y-coordinate when the x-coordinate is 0. Substitute \( x = 0 \) into the equation:\[y = \frac{1}{3}(0) + 1\]This simplifies to: \[y = 0 + 1 = 1\].Thus, the ordered pair is \((0, 1)\).

Step 3: Determine y for x = 3

Next, find the y-coordinate when the x-coordinate is 3. Substitute \( x = 3 \) into the equation:\[y = \frac{1}{3}(3) + 1\]This simplifies to: \[y = 1 + 1 = 2\].So, the ordered pair is \((3, 2)\).

Step 4: Graph the Equation

With the ordered pairs \((-3, 0)\), \((0, 1)\), and \((3, 2)\), plot these points on a coordinate plane. Connect these points with a straight line, as it represents the linear equation \( y = \frac{1}{3}x + 1 \). The line should pass through all the points, reflecting a slope of \( \frac{1}{3} \) and a y-intercept at 1.

Key Concepts

Graphing Linear Equations

Graphing linear equations is all about representing mathematical relationships visually. When you have an equation in the form of \( y = mx + b \), your goal is to find points that satisfy this equation and then plot these points on a coordinate plane. You will see the points line up in a straight line when connected, reflecting the linear nature of the equation.

• Start by identifying several ordered pairs \((x, y)\) that satisfy your equation.
• Substitute different values for \(x\) to find corresponding \(y\) values.
• Plot these points on the graph.
• Use a ruler to draw a line through the points. Extend the line in both directions using arrows to show it continues infinitely.

Graphing linear equations not only helps clarify the solutions visually but also makes it easier to understand how changes in \(x\) affect \(y\).

Coordinate Plane

The coordinate plane is a two-dimensional space where we plot points to visualize algebraic relationships. It's made of two perpendicular lines, or axes, often labeled \(x\) (horizontal) and \(y\) (vertical).

• The intersection of these lines is called the origin, marked as \((0, 0)\).
• The axes divide the plane into four sections called quadrants.
• The coordinate plane allows for visualization of equations, making it easier to see patterns and trends.
• To plot a point, you move horizontally from the origin on the \(x\)-axis and vertically on the \(y\)-axis.

The coordinate plane is an essential tool in mathematics because it provides a visual framework to better understand equations and their solutions.

Ordered Pairs

Ordered pairs are key to plotting points on the coordinate plane. Each ordered pair consists of two elements: \((x, y)\).

• The first number, \(x\), represents the horizontal position on the \(x\)-axis.
• The second number, \(y\), indicates the vertical position on the \(y\)-axis.
• Used together, ordered pairs specify precise locations on the coordinate plane.
• When graphing a linear equation, calculate ordered pairs to find specific points on the line the equation represents.

Ordered pairs are a fundamental concept in mathematics, crucial for graphing equations, solving systems of equations, and understanding relationships between variables.

Slope and Intercept

When graphing a linear equation, two important features to identify are the slope and the y-intercept.

• The **slope**, often represented by \(m\), shows how steep the line is. It indicates how much \(y\) changes for a change in \(x\).
• The **y-intercept** is the point where the line crosses the y-axis, denoted by \(b\) in the equation \(y = mx + b\).
• The slope can be found by dividing the difference in y-values by the difference in x-values for any two points on the line: \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
• The y-intercept is simply the y-value when \(x = 0\).

Knowing the slope and intercept helps in quickly graphing a line and understanding the relationship between the variables in the equation.