The graph of a system of linear equations in two variables which shows only one solution
Introduction to Systems of EquationsA system of equations consists of two or more equations with two or more variables, where any solution must satisfy all of the equations in the system at the same time. Show
Learning Objectives Explain what systems of equations can represent Key TakeawaysKey Points
Key Terms
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 of the system's equations at the same time. Some linear systems may not have a solution, while 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. 2x+y=153x−y=52x + y = 15 \\ 3x - y = 5 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. 2(4)+7=153(4)−7=52(4) + 7 = 15 \\ 3(4) - 7 = 5 Both of these statements are true, so (4,7)(4, 7) is indeed a solution to the system of equations. 3x+2y−z=12x−2y+4z=−2−x+12y−z=03x + 2y - z = 12 \\ x - 2y + 4z = -2 \\ -x + 12y -z = 0 is a system of three equations in the three variables x,y,zx, y, z . A solution to the system above is given by x=1y=−2z=−2x = 1 \\ y = -2 \\ z = - 2 since it makes all three equations valid. Types of Linear Systems and Their SolutionsIn general, a linear system may behave in any one of three possible ways:
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.An independent system has exactly one solution pair (x,y)(x, y) . 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. Solving Systems GraphicallyA 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. Learning Objectives Solve a system of equations in two variables graphically Key TakeawaysKey Points
Key Terms
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:
Here, we will address the graphical method. Solving Systems GraphicallySome 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.Converting to Slope-Intercept FormBefore successfully solving a system graphically, one must understand how to graph equations written in standard form, or A x+By=CAx+By=C . 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. y=mx+by=mx+b , where m = slope and b = y-intercept. Ax+By=CB y=−Ax+Cy=−Ax+CB y=−ABx+CB\begin{align} \displaystyle Ax+By&=C \\By&=-Ax+C \\y&=\frac{-Ax+C}{B} \\y&=-\frac{A}{B}x+\frac{C}{B} \end{align} Now −AB\displaystyle -\frac{A}{B} is the slope m, and CB\displaystyle \frac{C}{B} is the y-intercept b. Identifying Solutions on a GraphOnce 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 MethodThe substitution method is a way of solving a system of equations by expressing the equations in terms of only one variable. Learning Objectives Solve systems of equations in two variables using substitution
Key TakeawaysKey Points
Key Terms
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.
Solving with the Substitution MethodLet's practice this by solving the following system of equations: x−y=−1x-y=-1 x+2y=−4x+2y=-4 We begin by solving the first equation so we can express x in terms of y. x−y=−1x=y−1\begin{align} \displaystyle x-y&=-1 \\x&=y-1 \end{align} Next, we will substitute our new definition of x into the second equation: x+2y=−4(y−1)+2y=−4\displaystyle \begin{align} x+2y&=-4 \\(y-1)+2y&=-4 \end{align} Note that now this equation only has one variable (y). We can then simplify this equation and solve for y: (y−1)+2 y=−43y−1=−43y=−3y=−1 \displaystyle \begin{align} (y-1)+2y&=-4 \\3y-1&=-4 \\3y&=-3 \\y&=-1 \end{align} 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. x−y=−1x− (−1)=−1x+1=−1 x=−1−1x=−2 \displaystyle \begin{align} x-y&=-1 \\x-(-1)&=-1 \\x+1&=-1 \\x&=-1-1 \\x&=-2 \end{align} Thus, the solution to the system is: (−2,−1) (-2, -1) , which is the point where the two functions graphically intersect. Check the solution by substituting the values into one of the equations. x−y=−1(−2)−(−1) =−1−2+1=−1 −1=−1\displaystyle \begin{align} x-y&=-1 \\(-2)-(-1)&=-1 \\-2+1&=-1 \\-1&=-1 \end{align} The Elimination MethodThe elimination method is used to eliminate a variable in order to more simply solve for the remaining variable(s) in a system of equations. Learning Objectives Solve systems of equations in two variables using elimination Key TakeawaysKey Points
Key Terms
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.
Solving with the Elimination MethodThe elimination method can be demonstrated by using a simple example: 4x+y=82y+x=9\displaystyle 4x+y=8 \\ 2y+x=9 First, line up the variables so that the equations can be easily added together in a later step: 4x+y=8x+2y=9\displaystyle \begin{align} 4x+y&=8 \\x+2y&=9 \end{align} 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 example, the variable y can be eliminated if we multiply the top equation by −2 -2 and then add the equations together. −2(4x+y=8)x+2y=9 \displaystyle \begin{align} -2(4x+y&=8) \\x+2y&=9 \end{align} Result: −8x−2y=−16x+2y=9 \displaystyle \begin{align} -8x-2y&=-16 \\x+2y&=9 \end{align} Now add the equations to eliminate the variable y. −8x+x−2y+2y=−16+9−7x =−7\displaystyle \begin{align} -8x+x-2y+2y&=-16+9 \\-7x&=-7 \end{align} Finally, solve for the variable x. −7x=−7x =−7−7x=1\displaystyle \begin{align} -7x&=-7 \\x&=\frac{-7}{-7} \\x&=1 \end{align} Then go back to one of the original equations and substitute the value we found for x. It is easiest to pick the simplest equation, but either equation will work. 4x+y=84(1)+y=84+y=8y=4\displaystyle \begin{align} 4x+y&=8 \\4(1)+y&=8 \\4+y&=8 \\y&=4 \end{align} Therefore, the solution of the equation is (1,4). It is always important to check the answer by substituting both of these values in for their respective variables into one of the equations. 4x+y=84(1)+4=84+4=88=8\displaystyle \begin{align} 4x+y&=8 \\4(1)+4&=8 \\4+4&=8 \\8&=8 \end{align} Inconsistent and Dependent Systems in Two VariablesFor linear equations in two variables, inconsistent systems have no solution, while dependent systems have infinitely many solutions. Learning Objectives Explain when systems of equations in two variables are inconsistent or
dependent both graphically and algebraically. Key TakeawaysKey Points
Key Terms
Recall that a linear system may behave in any one of three possible ways:
Also recall that each of these possibilities corresponds to a type of system of linear equations in two variables. An independent system of equations has exactly one solution (x,y)(x,y) . An inconsistent system has no solution, and a dependent system has an infinite number of solutions. Dependent Systems The equations of a linear system are independent if none of the equations can be derived algebraically from the others. When the equations are independent, each equation contains new information about the variables, and removing any of the equations increases the size of the solution set.
Systems that are not independent are by definition dependent. Equations in a dependent system can be derived from one another; they describe the same line. They do not add new information about the variables, and the loss of an equation from a dependent system does not change the size of the solution set. 0=00 = 0 . 3x+2y=66x+4y=123x+2y = 6 \\ 6x+4y = 12 We can apply the elimination method to evaluate these. If we were to multiply the first equation by a factor of −2-2 , we would have: − 2(3x+2y=6)−6x−4y =−12\displaystyle \begin{align} -2(3x+2y&=6) \\-6x-4y&=-12 \end{align} Adding this to the second equation would yield 0=00=0 . Thus, the two lines are dependent. Also note that they are the same equation scaled by a factor of two; in other words, the second equation can be derived from the first. Dependent system: The equations 3x+2y=63x + 2y = 6 and 6x+4y=126x + 4y = 12 are dependent, and when graphed produce the same line. Note that there are an infinite number of solutions to a dependent system, and these solutions fall on the shared line. Inconsistent SystemsA linear system is consistent if it has a solution, and inconsistent otherwise. Recall that the graphical representation of an inconsistent system consists of parallel lines that have the same slope but different yy -intercepts. They will never intersect. 0=10 = 1 . 3x+2y=63x+2y= 123x+2y = 6 \\ 3x+2y = 12 We can apply the elimination method to attempt to solve this system. Subtracting the first equation from the second one, both variables are eliminated and we get 0=60 = 6 . This is a contradiction, and we are able to identify that this is an inconsistent system. The graphs of these equations on the xyxy -plane are a pair of parallel lines. Inconsistent system: The equations3x+2 y=63x + 2y = 6 and 3x+2y=123x + 2y = 12 are inconsistent. In general, inconsistencies occur if the left-hand sides of the equations in a system are linearly dependent, and the constant terms do not satisfy the dependence relation. A system of equations whose left-hand sides are linearly independent is always consistent. Applications of Systems of EquationsSystems of equations can be used to solve many real-life problems in which multiple constraints are used on the same variables. Learning Objectives Apply systems of equations in two variables to real world
examples Key TakeawaysKey Points
Key Terms
Systems of Equations in the Real WorldA system of equations, also known as simultaneous equations, is a set of equations that have multiple variables. The answer to a system of equations is a set of values that satisfies all equations in the system, and there can be many such answers for any given system. Answers are generally written in the form of an ordered pair: (x,y)\left( x,y \right) . Approaches to solving a system of equations include substitution and elimination as well as graphical techniques. Planning an Event A system of equations can be used to solve a planning problem where
there are multiple constraints to be taken into account: 5656 . Second, there must be one teacher for every seven students.
So, how many students and how many teachers are invited to the party? TT , and the number of students will be SS . 5656 , so: T+S=56T+S=56 For every seven students, there must be one teacher, so: S7=T\frac{S}{7}=T Now we have a system of equations that can be solved by substitution, elimination, or graphically. The solution to the system is S=49S=49 and T= 7T=7 . Finding Unknown Quantities This next example illustrates how systems of equations are used to find quantities. 7575 students and teachers are in a field, picking sweet potatoes for the needy. Kasey picks three times as many sweet potatoes as Davis—and then, on the way back to the car, she picks up five more! Looking at her newly increased pile, Davis remarks, "Wow, you've got 2929 more potatoes than I do!" How many sweet potatoes did Kasey and
Davis each pick? KK , and the number of sweet potatoes that Davis picks is DD . K−5=3DK-5 = 3D D+ 29=KD+29 = K From here, substitution, elimination, or graphing will reveal that K=41K=41 and D=12D=12 . Other ApplicationsThere are a multitude of other applications for systems of equations, such as figuring out which landscaper provides the best deal, how much different cell phone providers charge per minute, or comparing nutritional information in recipes. Licenses and AttributionsCC licensed content, Shared previously
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What is the graph of a system of linear equations that has one solution?A system of linear equations has one solution when the graphs intersect at a point. A coordinate plane. The x- and y-axes both scale by one-half.
What do you call the graph of linear equations in two variables which shows only one solution?The lines which intersect at a common point gives exactly one solution.These are called as intersecting lines. Answer : D.
What system of linear equations in two variables has only one solution?An independent system has exactly one solution pair (x,y) . The point where the two lines intersect is the only solution. An inconsistent system has no solution.
What do you call the graph of a system of linear equations in two variables which has no solution?When you graph the equations, both equations represent the same line. If a system has no solution, it is said to be inconsistent . The graphs of the lines do not intersect, so the graphs are parallel and there is no solution.
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