Foundational Mathematics for Corequisite Courses: Succeeding at College Mathematics

Section 12.1 Get it Straight

In Definition 4.1.4, we saw that solving an equation means to find the value (or values) of the variable (or variables) that make the equation true, and that solutions are the specific values of the variables that accomplish that. We are going to take another look at those ideas, but in the context of using multiple variables simultaneously. Specifically, we are going to be working with linear equations in the variables \(x\) and \(y\text{.}\)

Long lists of ordered pairs are not the most intuitive way to present solutions. For more than just a couple points, it makes sense to transition to using a chart, such as the one in the margin. Sometimes a chart like this can help us see a pattern in the numbers if the pattern is simple. But even with that, charts have limited value because it's still just a list of numbers. It would be better to use a more visual representation of this information.

Mathematicians often use a coordinate plane to represent solutions to two variable equations. The coordinate plane is a picture where specific positions represent specific ordered pairs. Most students are familiar with the basic design of the standard rectangular coordinate grid, which is the grid we will be using.

Here is a quick reminder of some of the basic terminology:

• The labeled horizontal line is called the \(x\)-axis

• The labeled vertical line is called the \(y\)-axis

• The intersection of those two lines is called the origin.

• The quadrants are numbered with capital roman numerals, starting with I on the upper right and working around in a counter-clockwise manner.

Students often learn coordinates as a sequence of motions starting from the origin. The point \((2,3)\) is located by starting from the origin, moving right 2, and then moving up 3. Negative \(x\)-coordinates correspond to moving to the left instead of to the right, and negative \(y\)-coordinates correspond to moving down instead of up.

But there is another way to look at this which is a little bit more general. Rather than thinking about this in terms of movement, we can think about this in terms of the intersection of two lines. Lines of the form \(x = \text{(Number)}\) are vertical lines corresponding to the coordinates on the \(x\)-axis and lines of the form \(y = \text{(Number)}\) are horizontal lines corresponding to the coordinates on the \(y\)-axis. It is the overlap of these lines that creates the coordinate grid.

Once you have this, then you can see that the positions are actually the intersection of two of these lines, corresponding to the specific value and the specific value. Here are the two ways of interpreting the point \((x,y) = (2,3)\) visualized side-by-side.

Activity 12.1.2. Plotting Points.

Part of mathematical thinking is the ability to conceptualize the same result in multiple ways. Locating points on a coordinate grid is an example of this. We can think of it both in terms of movement and in terms of the intersection of lines.

Try it!

Plot the point and draw a visualization for both conceptualizations of locating the point.

Solution.

We can generalize the idea of giving locations as the intersection of two lines by allowing ourselves to use curves. For example, locations on the earth are found as the intersection of the latitude and longitude lines (which are actually curves on the globe). Your location in a city can often be described as being near the intersection of two streets (which may not be straight). In trigonometry, there's another coordinate grid that's built around circles and lines pointing out from the origin called polar coordinates.

Once we have the ability to locate points on the coordinate plane, we can then plot lots of points on the same coordinate grid and look for a pattern.

Activity 12.1.3. Plotting Solutions of Linear Equations.

We saw earlier that we can generate solutions to a linear equation by inspection. If we plot those solutions on a grid, they will all appear in a line, which is why we call the equation a linear equation. Here is an example of solving the equation \(x - 2y = -1\text{.}\) After plotting the points, we can draw in the shape that is implied by the points.

Try it!

Find four solutions of the equation \(3x - 2y = -1\text{.}\) Plot the points and sketch the solution.

Solution.

An important feature to recognize is that the line that is drawn represents all of the solutions. It turns out that every single point on the line will solve the equation, even the ones that fall in between the grid points. This is where all those decimal solutions can be found.

However, while the sketch of the line is an important tool for building intuition, you need to be very careful about trying to guess exact values on the basis of a sketch. If you're reading values from a graph that are not on the grid lines, you must always acknowledge that you are only giving an approximation.