Problem 27 - Prealgebra

Question

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

Step-by-Step Solution

Answer

Three solutions are: (-1, -2), (0, 0), and (1, 2). Plot these points and draw a straight line to graph the equation.

Step 1: Understanding the Equation

The given equation is a linear equation in the form of \(y = mx + c\). Here, \(m = 2\) and \(c = 0\), which indicates that the line passes through the origin and has a slope of 2.

Step 2: Choosing Values for x

To find solutions, choose some arbitrary values for \(x\). For simplicity, we'll start with \(x = -1, 0,\) and \(1\).

Step 3: Calculating y for x = -1

Substitute \(x = -1\) into the equation \(y = 2x\): \[y = 2(-1) = -2\] So, the point is \((-1, -2)\).

Step 4: Calculating y for x = 0

Substitute \(x = 0\) into the equation \(y = 2x\): \[y = 2(0) = 0\] So, the point is \((0, 0)\).

Step 5: Calculating y for x = 1

Substitute \(x = 1\) into the equation \(y = 2x\): \[y = 2(1) = 2\] So, the point is \((1, 2)\).

Step 6: Plotting the Points

On a graph, plot the points \((-1, -2)\), \((0, 0)\), and \((1, 2)\). These points should lie on a straight line.

Step 7: Sketching the Graph

Draw a line through the points you plotted. This line represents the graph of the equation \(y = 2x\).

Key Concepts

Graphing

Graphing linear equations helps us visualize the relationship between variables. To graph an equation like \(y = 2x\), we begin by creating a set of coordinate points. Each point represents a specific \(x\) value and its corresponding \(y\) value computed from the equation.

• We first select several \(x\) values, such as \(-1, 0,\) and \(1\).
• We plug these \(x\) values into the equation to find \(y\).
• The resulting \(x, y\) pairs, such as \((-1, -2)\), \((0, 0)\), and \((1, 2)\), become points on the graph.

Plotting these points on a coordinate plane and connecting them with a straight line makes the equation's graph visible. The line passes through all points, illustrating the equation’s linear nature.

Visually, graphing makes it easier to see how changes in \(x\) affect \(y\). It offers insights into the equation that numbers alone might conceivably conceal.

Slope

Slope is a crucial concept when understanding linear equations and their graphs. In the equation \(y = mx + c\), the slope is represented by the coefficient \(m\). It describes the line's steepness and direction.

In our example, the slope \(m = 2\). This value means for every 1-unit increase in \(x\), \(y\) increases by 2 units. Here's how slope is symbolically understood:

• If \(m > 0\), the line rises as it moves from left to right (positive slope).
• If \(m < 0\), the line falls from left to right (negative slope).
• If \(m = 0\), the line is horizontal, indicating no rise or fall as \(x\) changes.

The slope shows the rate of change. A greater absolute value of \(m\) depicts a steeper line. Knowing the slope helps quickly predict how the graph moves without recalculating each point.

Coordinate Points

To plot linear equations, understanding coordinate points is key. A coordinate point is a pair of numbers \((x, y)\) that pinpoints a location on a graph.

Here's a closer look at the essential traits:

• The first number, \(x\), dictates the horizontal position on the graph.
• The second number, \(y\), indicates the vertical position.
• Together, they provide a complete location in the two-dimensional plane.

In our example, the points \((-1, -2)\), \((0, 0)\), and \((1, 2)\) reflect the line made by \(y = 2x\). Without coordinate points, visualizing and plotting equations would become complicated. They are foundational in graphing, ensuring each point is accurately positioned, which aids in drawing the correct line and interpreting data.