Problem 25 - Prealgebra

Question

Find three solutions to each of the equations and use them to draw the graph. (GRAPH CANT COPY) $$y=2 x+3$$

Step-by-Step Solution

Answer

The points (-1, 1), (0, 3), and (1, 5) satisfy the equation y = 2x + 3.

Step 1: Choose Values for x

To find three solutions to the equation, we start by choosing different values for \(x\). This will allow us to calculate corresponding \(y\) values and plot these points on a graph. Let's choose \(x = -1\), \(x = 0\), and \(x = 1\).

Step 2: Calculate y for x = -1

Substitute \(x = -1\) into the equation \(y = 2x + 3\):\[ y = 2(-1) + 3 = -2 + 3 = 1 \]For \(x = -1\), \(y = 1\). This gives us the point \((-1, 1)\).

Step 3: Calculate y for x = 0

Substitute \(x = 0\) into the equation:\[ y = 2(0) + 3 = 0 + 3 = 3 \]For \(x = 0\), \(y = 3\). This gives us the point \((0, 3)\).

Step 4: Calculate y for x = 1

Substitute \(x = 1\) into the equation:\[ y = 2(1) + 3 = 2 + 3 = 5 \]For \(x = 1\), \(y = 5\). This gives us the point \((1, 5)\).

Step 5: Plotting the Points

Now we have three points: \((-1, 1)\), \((0, 3)\), and \((1, 5)\). These points can be plotted on the Cartesian plane. This forms a straight line because the equation \(y = 2x + 3\) is a linear equation.

Key Concepts

Linear Equation

A linear equation is an equation that gives you a straight line when you graph it on a coordinate plane. Its general form is \( y = mx + b \), where \( m \) represents the slope, and \( b \) represents the y-intercept. This type of equation only has variables raised to the first power, meaning there are no squares or square roots involved. Understanding what a linear equation is can help you visualize how the graph will look even before plotting any points. In our case, the given equation \( y = 2x + 3 \) is linear because it has the form \( y = mx + b \). This tells us that when we graph it, it'll form a straight line, allowing us to predict the relationship between \( x \) and \( y \) values.

Coordinate Plane

The coordinate plane is a two-dimensional surface where you can plot points to represent mathematical equations. It consists of two perpendicular lines: one horizontal called the x-axis, and the other vertical called the y-axis. Each point on this plane is identified by a pair of numbers \( x, y \), known as coordinates.

• The x-coordinate tells you how far to move horizontally.
• The y-coordinate tells you how far to move vertically.

To plot a point, you start from the origin, which is the center of the plane where the x-axis and y-axis intersect. This is marked as \( (0,0) \). Moving to the right or up from the origin increases the value of the x or y coordinates, respectively, while moving to the left or down decreases the values.

Slope-Intercept Form

The slope-intercept form of a linear equation is written as \( y = mx + b \). Knowing how to interpret this form helps you understand the behavior of the line on a graph:

• **Slope** (\( m \)): It indicates the steepness or incline of the line. In our example \( y = 2x + 3 \), the slope \( m = 2 \) tells us that for every increase of 1 in \( x \), \( y \) increases by 2.
• **Y-intercept** (\( b \)): It is the point where the line crosses the y-axis. For \( y = 2x + 3 \), the y-intercept \( b = 3 \) means the line crosses the y-axis at point (0, 3).

Understanding the slope-intercept form is crucial for quickly graphing or interpreting a linear equation without needing to compute numerous points.

Plotting Points

Plotting points is a straightforward concept but essential for visualizing equations on a graph. After finding your x and y values from the linear equation, you can place these points on the coordinate plane.

• First, take your x-value and start at the origin. Move this many units left or right on the x-axis, depending on whether the x-value is negative or positive.
• Next, from this position, move up or down based on your y-value. Again, whether you go up or down depends on if the y-value is positive or negative.

Once you've plotted all the points, you connect them with a straight line, which represents your linear equation graphically. In our example, after plotting the points \((-1, 1)\), \((0, 3)\), and \((1, 5)\), we can clearly see that they align straight, confirming that the equation and plot form a linear relationship.