Problem 14 - Algebra 1: Concepts and Skills

Question

Use a table of values to graph the equation. $$ y=x+5 $$

Step-by-Step Solution

Answer

The points that will be graphed are (-2,3), (0,5) and (2,7). Draw a line that passes through all these points to give the graph of \(y=x+5\).

Step 1: Choose values for x

First, choose some values for \(x\) which will be used to find the corresponding \(y\) values using the equation \(y = x + 5\). For simplicity, let's choose the values -2, 0 and 2.

Step 2: Calculate the corresponding y values

Now, replace \(x\) in the equation \(y=x+5\) with the chosen \(x\) values and calculate the corresponding \(y\) values. When \(x=-2\), \(y=-2+5=3\). When \(x=0\), \(y=0+5=5\). When \(x=2\), \(y=2+5=7\). This gives us three points (-2,3), (0,5) and (2,7) to plot.

Step 3: Plot the points and draw the line

Finally, plot these points on the graph. Then draw a straight line that passes through these points to graph the equation \(y=x+5\).

Key Concepts

Table of Values

The 'table of values' is an essential tool for solving and graphing linear equations. It helps to visualize the relationship between two variables, typically denoted as 'x' and 'y'. To create a table of values, you select specific numbers to substitute for 'x', and then compute the corresponding 'y' values using the equation provided.

For instance, if we consider the linear equation y=x+5, we can choose a series of x-values, such as -2, 0, and 2. We then apply these x-values to the equation to find y. When x=-2, we find y=3; for x=0, we calculate y=5; and for x=2, y equates to 7. These computations neatly fill our table, resulting in pairs (-2,3), (0,5), and (2,7). These are then used as coordinates to plot on a graph. This methodical approach clarifies the linear relationship and allows for a seamless transition to graphing the equation.

Linear Equation

A 'linear equation' represents a straight line when graphed on a coordinate plane. The general form is y=mx+b, where m indicates the slope (the steepness of the line) and b refers to the y-intercept (where the line crosses the y-axis). The equation y=x+5 is a simple linear equation where the slope (m) is 1 and the y-intercept (b) is 5.

In this specific equation, for every unit increase in 'x', 'y' increases by the same amount. This indicates a slope of 1, which implies a 45-degree angle if the x and y axes are scaled equally. Understanding the components of a linear equation helps to predict the shape and position of the line before even plotting it on a graph, providing insight into the relationship between the variables.

Plotting Points

Placing points on a coordinate graph, commonly referred to as 'plotting points', transforms numerical relationships into visual ones. To plot a point, you locate the x-value on the horizontal axis and the y-value on the vertical axis. The intersection of these values on the graph represents the specific point.

Once you have computed your table of values for the equation y=x+5, you can begin plotting. The pairs (-2,3), (0,5), and (2,7) correspond with specific spots on the graph. Starting with the point (-2,3), you move left from the origin because the x-value is negative and up because the y-value is positive. This systematic plotting for each pair leads to a series of dots on your graph that should form a straight line when connected. Through this method, mathematical equations are made visual and more intuitive to understand.