How many variables are there in a linear equation

A skateboard manufacturer introduces a new line of boards. The manufacturer tracks its costs, which is the amount it spends to produce the boards, and its revenue, which is the amount it earns through sales of its boards. How can the company determine if it is making a profit with its new line?

How many skateboards must be produced and sold before a profit is possible? In this section we will consider linear equations with two variables to answer these and similar questions. In order to investigate situations such as that of the skateboard manufacturer, we need to recognize that we are dealing with more than one variable and likely more than one equation.

A system of linear equations consists of two or more linear equations made up of two or more variables such that all equations in the system are considered simultaneously. To find the unique solution to a system of linear equations, we must find a numerical value for each variable in the system that will satisfy all equations in the system at the same time.

Some linear systems may not have a solution and others may have an infinite number of solutions. In order for a linear system to have a unique solution, there must be at least as many equations as there are variables.

Even so, this does not guarantee a unique solution. In this section, we will look at systems of linear equations in two variables, which consist of two equations that contain two different variables. For example, consider the following system of linear equations in two variables. The solution to a system of linear equations in two variables is any ordered pair that satisfies each equation independently. We can verify the solution by substituting the values into each equation to see if the ordered pair satisfies both equations.

Shortly we will investigate methods of finding such a solution if it exists. In addition to considering the number of equations and variables, we can categorize systems of linear equations by the number of solutions. A consistent system of equations has at least one solution. A consistent system is considered to be an independent system if it has a single solution, such as the example we just explored.

The two lines have different slopes and intersect at one point in the plane. A consistent system is considered to be a dependent system if the equations have the same slope and the same y -intercepts. In other words, the lines coincide so the equations represent the same line. Every point on the line represents a coordinate pair that satisfies the system.

Thus, there are an infinite number of solutions. Another type of system of linear equations is an inconsistent system , which is one in which the equations represent two parallel lines.

The lines have the same slope and different y- intercepts. There are no points common to both lines; hence, there is no solution to the system. There are three types of systems of linear equations in two variables, and three types of solutions. We can see the solution clearly by plotting the graph of each equation. Since the solution is an ordered pair that satisfies both equations, it is a point on both of the lines and thus the point of intersection of the two lines.

There are multiple methods of solving systems of linear equations. For a system of linear equations in two variables, we can determine both the type of system and the solution by graphing the system of equations on the same set of axes. We can check to make sure that this is the solution to the system by substituting the ordered pair into both equations. Yes, in both cases we can still graph the system to determine the type of system and solution.

If the two lines are parallel, the system has no solution and is inconsistent. If the two lines are identical, the system has infinite solutions and is a dependent system.

Plot the three different systems with an online graphing tool. Categorize each solution as either consistent or inconsistent. If the system is consistent determine whether it is dependent or independent. You may find it easier to plot each system individually, then clear out your entries before you plot the next. Solving a linear system in two variables by graphing works well when the solution consists of integer values, but if our solution contains decimals or fractions, it is not the most precise method.

We will consider two more methods of solving a system of linear equations that are more precise than graphing. One such method is solving a system of equations by the substitution method , in which we solve one of the equations for one variable and then substitute the result into the second equation to solve for the second variable. Recall that we can solve for only one variable at a time, which is the reason the substitution method is both valuable and practical.

Yes, but the method works best if one of the equations contains a coefficient of 1 or —1 so that we do not have to deal with fractions. We present three different examples, and also use a graphing tool to help summarize the solution for each example.

A third method of solving systems of linear equations is the addition method, this method is also called the elimination method. In this method, we add two terms with the same variable, but opposite coefficients, so that the sum is zero. Of course, not all systems are set up with the two terms of one variable having opposite coefficients.

Often we must adjust one or both of the equations by multiplication so that one variable will be eliminated by addition. Both equations are already set equal to a constant. We gain an important perspective on systems of equations by looking at the graphical representation.

See the graph below to find that the equations intersect at the solution. We do not need to ask whether there may be a second solution because observing the graph confirms that the system has exactly one solution.

Adding these equations as presented will not eliminate a variable. Check the solution in the original second equation. First clear each equation of fractions by multiplying both sides of the equation by the least common denominator.

A linear equation in two variables can be described as a linear relationship between x and y , that is, two variables in which the value of one of them usually y depends on the value of the other one usually x. In this case, x is the independent variable, and y depends on it, so y is called the dependent variable. Whether or not it's labeled x , the independent variable is usually plotted along the horizontal axis. Most linear equations are functions. In other words, for every value of x , there is only one corresponding value of y.

When you assign a value to the independent variable, x , you can compute the value of the dependent variable, y. You can then plot the points named by each x , y pair on a coordinate grid. Students should already know that any two points determine a line. So graphing a linear equation in fact only requires finding two pairs of values and drawing a line through the points they describe. All other points on the line will provide values for x and y that satisfy the equation.

The graphs of linear equations are always lines. However, it is important to remember that not every point on the line that the equation describes will necessarily be a solution to the problem that the equation describes. For example, the problem may not make sense for negative numbers say, if the independent variable is time or very large numbers say, numbers over if the dependent variable is grade in class.

In this equation, for any given steady rate, the relationship between distance and time will be linear. However, distance is usually expressed as a positive number, so most graphs of this relationship will only show points in the first quadrant.

Notice that the direction of the line in the graph below is from bottom left to top right. Lines that tend in this direction have positive slope. A positive slope indicates that the values on both axes are increasing from left to right.

In this equation, since you won't ever have a negative amount of water in the bucket, the graph will show points only in the first quadrant. Notice that the direction of the line in this graph is top left to bottom right. Lines that tend in this direction have negative slope. A negative slope indicates that the values on the y- axis are decreasing as the values on the x- axis are increasing.

Again in this graph, we are relating values that only make sense if they are positive, so we show points only in the first quadrant. Moreover, in this case, since no polygon has fewer than 3 sides or angles and the number of sides or angles of a polygon must be a whole number, we show the graph starting at 3,3 and indicate with a dashed line that points between those plotted are not relevant to the problem. Since it's perfectly reasonable to have both positive and negative temperatures, we plot the points on this graph on the full coordinate grid.

The slope of a line tells two things: how steep the line is with respect to the y- axis and whether the line slopes up or down when you look at it from left to right. More technically speaking, slope tells you the rate at which the dependent variable is changing with respect to the change in the independent variable. Pick any two points on the line. To find how fast y is changing, subtract the y value of the first point from the y value of the second point: y 2 — y 1.

To find how fast x is changing, subtract the x value of the first point from the x value of the second point: x 2 — x 1. It does not matter which points along the line you designate as A and B , just as long as we're consistent with which is the "first" point x 1 , y 1 and which is the "second" x 2 , y 2. It is also the same value you will get if you choose any other pair of points on the line to compute slope.

The equation of a line can be written in a form that makes the slope obvious and allows you to draw the line without any computation. If students are comfortable with solving a simple two-step linear equation, they can write linear equations in slope-intercept form. In the equation, x and y are the variables.

The numbers m and b give the slope of the line m and the value of y when x is 0 b. The value of y when x is 0 is called the y -intercept because 0, y is the point at which the line crosses the y -axis.

You can draw the line for an equation matching this linear formula by plotting 0, b , then using m to find another point. Now look at b in the equation: —3 should be where the line intercepts the y -axis, and it is. When a line slopes up from left to right, it has a positive slope. This means that a positive change in y is associated with a positive change in x. The steeper the slope, the greater the rate of change in y in relation to the change in x. When the line represents real-world data points plotted on a coordinate plane, a positive slope indicates a positive correlation, and the steeper the slope, the stronger the positive correlation.

Consider a linear equation where the independent variable g is gallons of gas used and the dependent variable d is the distance traveled in miles. If you drive a big, old car, you get poor gas mileage. The amount of miles traveled is low relative to the amount of gas consumed, so the value m is a low number.

The slope of the line is fairly gradual. If instead you drive a light, efficient car, you get better gas mileage. You travel more miles relative to the same amount of gas consumed, so the value of m is greater and the line is steeper.

Both rates are positive because you still travel a positive number of miles for every gallon of gas you consume. When a line slopes down from left to right, it has a negative slope. This means that a negative change in y is associated with a positive change in x.

When the line represents real-world data points plotted on a coordinate plane, a negative slope indicates a negative correlation, and the steeper the slope, the stronger the negative correlation. Consider a line that represents the number of peppers left to plant after minutes spent gardening. Check the solution by substituting into both equations.

Can the substitution method be used to solve any linear system in two variables? Yes, but the method works best if one of the equations contains a coefficient of 1 or —1 so that we do not have to deal with fractions. A third method of solving systems of linear equations is the addition method. In this method, we add two terms with the same variable, but opposite coefficients, so that the sum is zero.

Of course, not all systems are set up with the two terms of one variable having opposite coefficients. Often we must adjust one or both of the equations by multiplication so that one variable will be eliminated by addition. Given a system of equations, solve using the addition method. Both equations are already set equal to a constant. Notice that the coefficient of in the second equation, —1, is the opposite of the coefficient of in the first equation, 1. We can add the two equations to eliminate without needing to multiply by a constant.

Now that we have eliminated we can solve the resulting equation for. Then, we substitute this value for into one of the original equations and solve for. The solution to this system is.

We gain an important perspective on systems of equations by looking at the graphical representation. See Figure to find that the equations intersect at the solution. We do not need to ask whether there may be a second solution because observing the graph confirms that the system has exactly one solution. Solve the given system of equations by the addition method.

Adding these equations as presented will not eliminate a variable. However, we see that the first equation has in it and the second equation has So if we multiply the second equation by the x -terms will add to zero. For the last step, we substitute into one of the original equations and solve for. Our solution is the ordered pair See Figure.

Check the solution in the original second equation. One equation has and the other has The least common multiple is so we will have to multiply both equations by a constant in order to eliminate one variable. Substitute into the original first equation. The solution is Check it in the other equation. First clear each equation of fractions by multiplying both sides of the equation by the least common denominator. Now multiply the second equation by so that we can eliminate the x -variable.

Add the two equations to eliminate the x -variable and solve the resulting equation. Substitute into the first equation. Now that we have several methods for solving systems of equations, we can use the methods to identify inconsistent systems. Recall that an inconsistent system consists of parallel lines that have the same slope but different -intercepts.

They will never intersect. When searching for a solution to an inconsistent system, we will come up with a false statement, such as. We can approach this problem in two ways. Because one equation is already solved for the most obvious step is to use substitution.

Clearly, this statement is a contradiction because Therefore, the system has no solution. The second approach would be to first manipulate the equations so that they are both in slope-intercept form.

We manipulate the first equation as follows. Comparing the equations, we see that they have the same slope but different y -intercepts. Therefore, the lines are parallel and do not intersect. Writing the equations in slope-intercept form confirms that the system is inconsistent because all lines will intersect eventually unless they are parallel. Parallel lines will never intersect; thus, the two lines have no points in common. The graphs of the equations in this example are shown in Figure.

Recall that a dependent system of equations in two variables is a system in which the two equations represent the same line. Dependent systems have an infinite number of solutions because all of the points on one line are also on the other line. After using substitution or addition, the resulting equation will be an identity, such as. Find a solution to the system of equations using the addition method.

With the addition method, we want to eliminate one of the variables by adding the equations. We can see that there will be an infinite number of solutions that satisfy both equations.

If we rewrote both equations in the slope-intercept form, we might know what the solution would look like before adding. Notice the results are the same. The general solution to the system is. The system is dependent so there are infinite solutions of the form.

Using what we have learned about systems of equations, we can return to the skateboard manufacturing problem at the beginning of the section. It can be represented by the equation where quantity and price. The revenue function is shown in orange in Figure. The cost function is the function used to calculate the costs of doing business. It includes fixed costs, such as rent and salaries, and variable costs, such as utilities. The cost function is shown in blue in Figure. The -axis represents quantity in hundreds of units.

The y -axis represents either cost or revenue in hundreds of dollars. The point at which the two lines intersect is called the break-even point.

In other words, the company breaks even if they produce and sell units. They neither make money nor lose money. The shaded region to the right of the break-even point represents quantities for which the company makes a profit. The shaded region to the left represents quantities for which the company suffers a loss.

The profit function is the revenue function minus the cost function, written as Clearly, knowing the quantity for which the cost equals the revenue is of great importance to businesses.

Given the cost function and the revenue function find the break-even point and the profit function. Write the system of equations using to replace function notation. Substitute the expression from the first equation into the second equation and solve for. Then, we substitute into either the cost function or the revenue function. The break-even point is. The profit function is found using the formula. To make a profit, the business must produce and sell more than 50, units. We see from the graph in Figure that the profit function has a negative value until when the graph crosses the x -axis.

Then, the graph emerges into positive y -values and continues on this path as the profit function is a straight line. This illustrates that the break-even point for businesses occurs when the profit function is 0.

The area to the left of the break-even point represents operating at a loss. The cost of a ticket to the circus is for children and for adults. On a certain day, attendance at the circus is and the total gate revenue is How many children and how many adults bought tickets?

The total number of people is We can use this to write an equation for the number of people at the circus that day. The revenue from all children can be found by multiplying by the number of children, The revenue from all adults can be found by multiplying by the number of adults, The total revenue is We can use this to write an equation for the revenue.

In the first equation, the coefficient of both variables is 1. We can quickly solve the first equation for either or We will solve for. Substitute the expression in the second equation for and solve for.

Substitute into the first equation to solve for. We find that children and adults bought tickets to the circus that day. Meal tickets at the circus cost for children and for adults.

If meal tickets were bought for a total of how many children and how many adults bought meal tickets? Access these online resources for additional instruction and practice with systems of linear equations. Can a system of linear equations have exactly two solutions? Explain why or why not. If you are solving a break-even analysis and get a negative break-even point, explain what this signifies for the company?