Check whether 2x 3y = 7 and 4x+6y 16 are consistent or inconsistent
Video transcriptUse elimination to solve for x and y. And they gave us two equations here-- x plus 2y is equal to 6 and 4x minus 2y is equal to 14. So to solve by elimination, what we do is we're going to add these two equations together so that one of the two variables essentially gets eliminated, gets canceled out. And what we could do right here, we see we have a plus 2y here, and we see we have a negative 2y right over here. So clearly if we added these two together, the y's would cancel out, and so that's exactly what we're going to do. We're going to add the left side of this equation to the left side of this equation, and the right-hand side of that equation to the right-hand side of the bottom equation. And just to make it clear that this should make sense is we're just using both of these constraints, whenever you learn about any type of equation. So if I have x plus 2y is equal to 6, you learned early on in algebra that you can manipulate this equation in any way as long as whatever you do to the left-hand side of the equation, you do the right-hand side. If this is equal to that, the only way that the equality will still hold is whatever you do to this-- whatever you add to this or multiply it by-- you also do to the right-hand side. So when we're adding these two equations, that's exactly what we're doing. We could say, hey, let's add 14 to both sides of this equation. So you could add 14 on this side, you could add 14 on that side. That wouldn't be anything new, but the second equation right here tells us that 4x minus 2y is the same thing as 14. So instead of adding 14 on the left-hand side, I could add 4x minus 2y. When we're adding these two equations, we're really just adding the same thing. You could view it as we're starting with this equation, and then we're adding the same thing to both sides. On the right-hand side it looks like we're adding 14 to the 6. On the left-hand side it looks like we're adding 4x minus 2y to whatever is on the left-hand side. But the second constraint tells us that 14 and 4x minus 2y are the same thing. So we're adding the same thing to both sides. So with that said, let's just do it. So the left-hand side, if we add it up, we have x plus 4x is 5x, and then the 2y cancels out with the negative 2y. And then on the right-hand side we have 6 plus 14. 6 plus 14 Is 20. So we're left with one equation with one unknown. 5x is equal to 20. We can divide both sides by 5, and we are left with x is equal to 4. Now we can go back and substitute in x equals 4 into either of these equations to solve for y. So let's use this top one. So we have 4 plus 2y is equal to 6. We can subtract 4 from both sides. So then we get 2y is equal to 2. Divide both sides by 2. We get y is equal to 1. So the solution, the x's and y's that satisfy both of these equations, are x is equal to 4, and y is equal to 1. So this is the solution for this system, or this coordinate would be the point of intersection of these two lines. And we can verify it. Let's verify that when we put x is equal to 4, and y is equal to 1, in this first equation it satisfies it. So we have 4 plus 2 times 1. That's 4 plus 2. That does, indeed, equal 6. And then the second equation right over here, you have 4 times 4 minus 2 times 1. This is equal to 16 minus 2, which does indeed equal 14. So it definitely does satisfy both of these equations. So we're done. X is equal to 4, and y is equal to 1. Solving Systems of Equations
Solving Systems of Equations There are three ways to solve a system of linear equations: graphing, substitution, and elimination. Graphing Method The solution to a system of linear equations is the ordered pair (or pairs) that satisfies all equations in the system. The solution is the ordered pair(s) common to all lines in the system when the lines are graphed. Lines that cross at a point (or points) are defined as a consistent system of equations. The place(s) where they cross are the solution(s) to the system. Parallel lines do not cross. They have the same slope and different y-intercepts. They are an example of an inconsistent system of equations. An inconsistent system of equations has no solution. Two equations that actually are the same line have an infinite number of solutions. This is an example of a dependent system of equations. Example Solve the system of equations graphically.
Solution Graph each line and determine where they cross. The lines intersect once at (2, −1). A graphic solution to a system of equations is only as accurate as the scale of the paper or precision of the lines. At times the point of intersection will need to be estimated on the graph. When an exact solution is necessary, the system should be solved algebraically, either by substitution or by elimination. Substitution Method To solve a system of equations by substitution, solve one of the equations for a variable, for example x. Then replace that variable in the other equation with the terms you deemed equal and solve for the other variable, y. The solution to the system of equations is always an ordered pair. Example Solve the following system of equations by substitution. x + 3y = 18 Solution Solve for a variable in either equation. (If possible, choose a variable that does not have a coefficient to avoid working with fractions.) In this case, it's easiest to rewrite the first equation by solving for x. x + 3y = 18 x = −3y + 18 Next, substitute (−3y + 18) in for x into the other equation. Solve for y.
Then, substitute y = 5 into your rewritten equation to find x.
Identify the solution. A check using x = 3 and y = 5 in both equations will show that the solution is the ordered pair (3, 5). Elimination Method 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. Example Find the solution to the system of equations by using the elimination method.
Solution Add the equations.
Isolate the variable in the new equation
Substitute x = 5 into either of the original equations to find y.
Identify the ordered pair that is the solution. A check in both equations will show that (5, −2) is a 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:
Both sides of the second equation above could be multiplied by −3. 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. Special Cases 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.
The equation is false. The system has no solution. If both variables drop out and the resulting expression is true, then the system is dependent and has infinite solutions.
The equation is true. The system has an infinite number of solutions. (Notice that both of the original equations reduce to 2x + 5y = 8. All solutions to the system lie on this line.) |
