Graphing Linear Equations
Now that we have solved equations in one variable, we will now work on solving equations in two variables and graphing equations on the coordinate plane. Graphs are very important for giving a visual representation of the relationship between two variables in an equation.
First, let's get acquainted with the coordinate plane (or Cartesian graph).
The coordinate plane was created by the French Mathematician Rene Descartes in order to goemetrically represent algebraic equations. This is often why the coordinate plane is called the Cartesian plane, or graph. When working with equations in more than one variable, using the Cartesian graph can be in important tool to make equations easier to visualize and understand.
The horizontal number line is the x-axis and the vertical number line is the y-axis. The point where both lines intersect is called the origin.
Each point on the graph is depicted by an ordered pair, where x is always the first value and y is always the second value in the ordered pair (x,y). This is because x is the independent variable, which means that it is the variable being changed. This makes y the dependent variable, which means that it is dependent on how x is being changed. We will explore this as we start to graph equations in terms of x and y. Now, even though there are two values in an order pair, they associate to only one point on the graph.
Let's plot the points A(0,0), B(1,2), C(-4,2), D(-3,-4), and E(4,-2).
Notice that A(0,0) is the origin because both it's x and y values are 0. For B(1,2), the x value would be 1 and the y value would be 2. To plot the point, we would go in the positive direction on the x axis until we hit 1, then we would go up on the positive y axis until we hit 2. This is where the point is located. We get our points by just lining up the x value and y value to get their locations, and we can do this for any coordinate pair.
Horizontal and Vertical Lines
On the coordinate plane, we know that each point must have an x and a y value. When we solved equations in one variable, it was easy to see that we had an x value. What we didn't realize is that we also had a y value as well. In fact, we had infinitely many y values. Similarly, if we were to solve a one variable equation in terms of y, we would have infinitely many x values. These equations do not form a point, but rather a horizontal or vertical line.
When x equals a number, y can take on any value and it would not change the equality. We could think of the equation as having a y value with 0 as a coefficient, so no matter what value y takes, it will always multiply by 0. This will form a vertical line.
Similarly, when y equals a number, x can take on any value and it would not affect the equality. We could think of the equation having value of 0x, so x can be any number and it would not affect the equation. This graph will be a horizontal line.
This makes sense, because the x axis is y = 0 and the y axis is x = 0.
Solutions of Equations in Two Variables
We have seen expressions and equations of one variable, mainly x. Here is an expression
When we plug in different values of x, we also yield a different output as well. Since this output varies depending on x, we can also use a variable to represent the output of x.
When dealing with equations in two variables, the solutions consist of x and y values that make the equation true when plugged into the equation. These solutions will turn out to be ordered pairs, and we will see that equations in 2 variables can have more than one solution, and often infinitely many solutions.
Given the equation
Determine whether the coordinates (1,5), (2,6), and (-1,1) are solutions to the equation.
Let's start with (1,5) and plug it into the equation for x and y.
This is true! This means that this point is a solution.
Let's try (2,6)
This is false because 6 does not equal 7, therefore it is not a solution.
Finally, let's plug in (-1,1).
This is another true statement, so (1,1) is a solution to the equation.
Let's plot both of the solutions we found to y = 2x+3 on the coordinate plane
These are not the only solutions to this equation. One method we could use to find other solutions to our equation is make a table of x and y values. We can do this by plugging in different x values and find their corresponding y values.
Now that we have a few coordinate points, let's plot them on the graph.
We can see that the points form a straight line, so we can draw a line through them. Any point on this line is a solution to the equation y = 2x+3. It is safe to say that the line we have drawn on the graph is the solution set to our equation.
For any two variable equation, we can attempt to graph the function by plugging in random x values to get our corresponding y values. This way, we have many points that we can graph. Some equations are easier to graph because they have noticeable patterns. We should keep in mind that most of the equations we work with will be in terms of x and y, because the coordinate plane is formed by the x and y axes.
Let's look at linear equations.
Linear equations are equations of two variables that form a line on the graph. A linear equation is defined where each term is either a constant or a product of a constant and a single variable. There are many different ways that linear equations can be represented algebraically and plotted graphically.
Here are different forms of Linear Equations
A linear equation in the form of two variables can be written in this form, where A, B, and C are constants. This form is beneficial because we can easily obtain the x and y intercepts by plugging in 0 for one of the variables. An intercept is the intersection of the line and either the x or y axis. We will see that these intercepts will help in plotting linear equations.
Write the following equation in standard form and plot the line on the graph.
First we multiply both sides by 3 to get rid of the fraction.
Then we subtract 2x from both sides to get the x and y on the same side
Let's rearrange it so our x value is first.
This is the standard form of our original equation. Since our original equation can be written in standard form, we know it is a linear equation (if an equation cannot be written in standard form, it is not linear).
Now let's plot the graph of the equation by finding our intercepts. First, let's find our y intercept by plugging in 0 for x.
We plugged in 0 for x and got -4 for y. Our coordinate would be (0,-4), which we call our y intercept. This is called our y intercept because it is the point where the graph of the equation intersects the y axis.
Let's find the x intercept by plugging in 0 for y.
When we plugged in y = 0, we got x = 6, so our coordinate is (6,0). This is the x intercept because it is the point where the graph crosses the x axis. Since we have our x and y intercepts and we know the equation is linear (we put it in Standard Form), we can graph the equation.
This line is the solution set of our equation. We should note that if we know an equation is linear, it only takes two points to construct the line on a graph. Just to make sure, it is always good to plot more than two points to check if the points are collinear (If they form a line). If we do not know it is linear, it is beneficial to plot a number of points to clearly see the curve of the graph. If we were given this graph without the algebraic representation, it would be hard to come up with the standard form of the equation, so we can use the following general forms of linear equations to find them.
This form is the most commonly used to represent linear equations. This form is the best way to find the slope and y intercept of a linear equation, where m is the slope and b is the y intercept.
Let's plot this equation using the slope-intercept form.
Comparing to our general slope-intercept equation, we can see that m = 2/3 and b = -4. Plotting this on a graph, we can obtain our line.
Since we have our y intercept and our slope, we can plot our y intercept and find other point on the line using the slope. Since m = 2/3, we can go up positive 2 and right positive 3 to obtain our next point on the line. We can repeat this process to get the line of our equation.
Finally, we have point-slope form. We can use the representation if we have any point on the line (it doesn't have to be an intercept) and the slope, or if we have any two points on the line.
Find the equation of the line through the point (3,-2) with slope m = 2/3. Let's plug the values into our equation.
We plug in (3,-2) for (x1,y1) and let m = 2/3
We have our equation! Now let's try it given two points.
Find the equation of the line through the points (-3,-6) and (3,-2). If we know the equation is linear, we can just plot the points and draw a line through them, but in this case we want to find the equation of the line. Let's plug them into the general point slope form and see what we get.
Since we don't know the slope but we have two points, we can plug our two points into the slope formula.
Converting Forms of Linear Equations
Even though we have three different forms of linear equations, they are all the same. The reason we have these different forms is because they are each beneficial for different geometric representations and ways of working with the information we have. The various forms of linear equations can be converted from one form to another.
(1) When converting from Standard form (Ax + By = C) to Slope-intercept form (y = mx+b), we have
(2) When converting from Standard form (Ax + By = C) to Point slope form [(y-y1) = m(x-x1)], we have
Finding the Equation of a Line
If we are given a graph of a line and we want to find its equation (or algebraic representation), we can find it a number of ways.
(1) Given two points on the line, we can plug them into the slope formula to find the slope and then use the point-slope form.
(2) Given any point on the line and the y intercept, we can plug them into the slope formula to find the slope and then use either the point-slope or slope-intercept form.
(3) Given any point on the line and the slope, use the point-slope form.
(4) Given the y intercept and the slope, use the slope-intercept form.
Plotting the Graph of a Linear Equation
Given any type of equation (it doesn't have to be linear), we can plug in a random x value and obtain a y value. We could plot points this way, but it is a tedious process and not completely necessary. Here are other ways of constructing the graph of a linear equation.
(1) Given an equation in any form, plug in any x value to the equation and find the y value. Plot the point on the graph and do this again for at least one more point. After we have two points, draw a line through the points for all solutions to the linear equation. If the points are noncollinear (they do not line up), then either the equation is not linear or there was an arithmetic mistake in finding them.
(2) Given the Standard Form of a linear equation (Ax + By = C), set the x value to 0 to find the y intercept and then the y value to 0 for the x intercept. Plot the points on the graph and draw a line through them.
(3) Given the Slope-Intercept Form (y = mx + b), plot the y intercept (0,b) and use the slope m to find the rest of the points on the graph.
(4) Given the Point-Slope Form [(y - y1) = m(x - x1)], plot the point (x1,y1) and use the slope m to find the rest of the points on the graph.
Systems of Linear Equations
When we graph more than one linear equation at once, we are considered to have a system of linear equations. Solving these systems will give us the point at which the lines intersect, which is quite relevant in various real life applications and is executed often in economics and higher level math.
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