Which point satisfies the system of equations

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.

Graph both equations on the same set of axes, use function notation so you can check your solution more easily later. You 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. In this section, we will focus primarily on systems of linear equations 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.

In this example, the ordered pair 4, 7 is the solution to the system of linear equations. We can verify the solution by substituting the values into each equation to see if the ordered pair satisfies both equations. Note that a system of linear equations may contain more than two equations, and more than two variables. For example,. A solution to the system above is given by. Each of these possibilities represents a certain type of system of linear equations in two variables. Each of these can be displayed graphically, as below.

Note that a solution to a system of linear equations is any point at which the lines intersect. Systems of Linear Equations: Graphical representations of the three types of systems.

The point where the two lines intersect is the only solution. An inconsistent system has no solution. Notice that the two lines are parallel and will never intersect. A dependent system has infinitely many solutions. The lines are exactly the same, so every coordinate pair on the line is a solution to both equations. A simple way to solve a system of equations is to look for the intersecting point or points of the equations.

This is the graphical method. A system of equations also known as simultaneous equations is a set of equations with multiple variables, solved when the values of all variables simultaneously satisfy all of the equations. The most common ways to solve a system of equations are:. Some systems have only one set of correct answers, while others have multiple sets that will satisfy all equations. Shown graphically, a set of equations solved with only one set of answers will have only have one point of intersection, as shown below.

This point is considered to be the solution of the system of equations. In a set of linear equations such as in the image below , there is only one solution. System of linear equations with two variables : This graph shows a system of equations with two variables and only one set of answers that satisfies both equations. A system with two sets of answers that will satisfy both equations has two points of intersection thus, two solutions of the system , as shown in the image below.

System of equations with multiple answers: This is an example of a system of equations shown graphically that has two sets of answers that will satisfy both equations in the system. You can always use a graphing calculator to represent the equations graphically, but it is useful to know how to represent such equations formulaically on your own.

The best way to convert an equation to slope-intercept form is to first isolate the y variable and then divide the right side by B , as shown below. Once you have converted the equations into slope-intercept form, you can graph the equations. To determine the solutions of the set of equations, identify the points of intersection between the graphed equations.

The ordered pair that represents the intersection s represents the solution s to the system of equations. The substitution method is a way of solving a system of equations by expressing the equations in terms of only one variable. The substitution method for solving systems of equations is a way to simplify the system of equations by expressing one variable in terms of another, thus removing one variable from an equation.

When the resulting simplified equation has only one variable to work with, the equation becomes solvable. Note that now this equation only has one variable y. We can then simplify this equation and solve for y :. Now that we know the value of y , we can use it to find the value of the other variable, x.

To do this, substitute the value of y into the first equation and solve for x. Check the solution by substituting the values into one of the equations. The elimination method is used to eliminate a variable in order to more simply solve for the remaining variable s in a system of equations. The elimination method for solving systems of equations, also known as elimination by addition , is a way to eliminate one of the variables in the system in order to more simply evaluate the remaining variable.

Once the values for the remaining variables have been found successfully, they are substituted into the original equation in order to find the correct value for the other variable. Next, look to see if any of the variables are already set up in such a way that adding them together will cancel them out of the system.

If not, multiply one equation by a number that allow the variables to cancel out. In this case, it's easiest to rewrite the first equation by solving for x. Identify the solution.

Another way to solve a system of equations is by using the elimination method. The aim of using the elimination method is to have one variable cancel out. The resulting sum will contain a single variable that can then be identified. Once one variable is found, it can be substituted into either of the original equations to find the other variable. Identify the ordered pair that is the solution. It may be necessary to multiply one or both of the equations in the system by a constant in order to obtain a variable that can be eliminated by addition.

For example, consider the system of equations below:. Multiplying the equation by the same number on both sides does not change the value of the equation. It will result in an equation whereby the x values can be eliminated through addition.

In some circumstances, both variables will drop out when adding the equations. If the resulting expression is not true, then the system is inconsistent and has no solution.