Problem 43 - Algebra 1: Concepts and Skills

Question

Use a table of values to graph the equation. $$ y-5 x=-2 $$

Step-by-Step Solution

Answer

The graph of the equation \(y = 5x - 2\) is a straight line that passes through the points (-1,-7), (0,-2), and (1,3).

Step 1: Understanding The Equation

First consider the equation \(y - 5x = -2\). Rewriting it in the standard form for linear equations, you get \(y = 5x - 2\). This equation describes a line with the slope 5 and y-intercept -2. This means that our graph should be a line which increases with the slope 5 and cuts the y-axis at -2.

Step 2: Creating A Table Of Values

We now create a table of values that includes a list of x-value and the corresponding y-values determined from the equation \(y = 5x - 2\) for each x-value.

Step 3: Computing Y-values

Choose several numbers for x and compute the respective y-values. For instance, you can pick x-values like -1, 0, and 1. Plugging these into our equation we get for x = -1, y = 5*(-1) - 2 = -7, for x = 0, y = 5*0 - 2 = -2, and for x = 1, y = 5*1 - 2 = 3. So our table now lists [(-1, -7), (0, -2), (1, 3)].

Step 4: Plotting the Points and Drawing the Line

Plot the points from the table into an xy-coordinate system. This gives us points at (-1,-7), (0,-2), and (1,3). Connect these points with a straight line, this is the graph of the equation.

Key Concepts

Table of Values

Imagine you are given a mysterious box that outputs a number whenever you input another number. The relationship between what you put in and what comes out can often be described using a graph in mathematics. A table of values functions similarly, acting as a tool to input numbers (called x-values) into an equation to see what we get out (called y-values).

For the equation in our exercise, written in slope-intercept form as \(y = 5x - 2\), we start by choosing some x-values. These can be any numbers, but often starting with integers like -1, 0, and 1 keeps it simple. Next, we calculate the corresponding y-values by replacing the x in the equation with the chosen numbers. For example, if x is 0, our y would be \(5(0) - 2 = -2\). Finally, we list these pairs of numbers in a two-column table, with 'x' on one side and 'y' on the other. This forms the basis for graphing the relationship on a coordinate plane.

Tables of values are fundamental, especially when starting out with linear equations because they allow us to see the pattern of the relationship between x and y, and help us plot the points to visualize the equation as a graph.

Slope-Intercept Form

If the mysterious box we mentioned earlier always gave out a number that was five times the input plus a little extra (or minus, if the extra is negative), you could express this consistent pattern with an equation. The slope-intercept form is one of the popular ways to write this and it looks like \(y = mx + b\), where 'm' represents the slope, and 'b' represents the y-intercept.

The key feature of the slope-intercept form is clarity. It directly tells you how steep the graph of the line is ('m') and where it strikes through the y-axis ('b'). In our equation, \(y = 5x - 2\), the number 5 is the slope. This tells you that for every step you go to the right along the x-axis, the value of y goes up by 5 steps. The number -2 is the y-intercept, indicating that if you climb up the y-axis, you would mark your beginning at -2 below the origin (where the x-axis and y-axis cross).

Students finding it tricky to remember the components of the slope-intercept form could think of 'm' as 'move' and 'b' as 'beginning'. So as you graph, you begin at 'b' and move as per 'm'.

Y-Intercept

The y-intercept is like the 'home base' for a line on a graph. It's the spot where the line hits the y-axis. To figure out where this is, just look for where the line crosses the y-axis. That point’s y-value is the y-intercept.

In our exercise, the y-intercept is -2, which is part of the equation \(y = 5x - 2\). It tells us that without moving left or right at all (which would mean our x-value is 0), our y-value starts off at -2. When graphing, you'd literally start plotting the line from this point. In real-life problems, this can mean many things, but it always points to an initial value before anything else starts changing. It could represent a starting balance, a base price before additional charges, or a point in time before an event begins.

The y-intercept provides an easy way to check if you’ve drawn the graph accurately. If your line doesn’t cross the y-axis at the correct y-intercept point, you’ll know to look over your calculations or plotting once more.