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Watch: Video Lesson Watch: Practice Set Watch: Test Solutions No Solution EquationAbout: Additional Resources: In this section, we learn about special case linear equations. When we encounter special case equations, we will see No Solution Equations and Equations that have infinitely many solutions. When solving equations, we will encounter three types of equations. These are conditional equations, identities, and contradictions. The first type of equation, known as a conditional equation is true under certain conditions, but false under others. As an example, suppose we look at 3x = 12. This equation is true when x = 4, but false when x is any other value. The second equation, an identity is always true, no matter what value replaces the variable. The left and the right side can be simplified to match each other. As an example, 3(x - 5) = 3x - 15. If we simplified each side we would get: 3x - 15 = 3x - 15. No matter what value we replace x with, the equation is true. For this type of equation, the solution is all real numbers. The last type of equation is known as a contradiction, which is also known as a No Solution Equation. This type of equation is never true, no matter what we replace the variable with. As an example, consider 3x + 5 = 3x - 5. This equation has no solution. There is no value that will ever satisfy this type of equation. + Show More + If you're seeing this message, it means we're having trouble loading external resources on our website. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Many students assume that all equations have solutions. This article will use three examples to show that assumption is incorrect.
Given the equation 5x - 2 + 3x = 3(x+4)-1 to solve, we will collect our like terms on the left hand side of the equal sign and distribute the 3 on the right hand side of the equal sign. 5x - 2 + 3x = 3(x+4)-1 is equivalent to 8x - 2 = 3x + 12 - 1, that is, 8x - 2 = 3x + 11. We will now collect all our x-terms on one side of the equal sign (it does not matter whether the x-terms are placed on the left side of the equal sign or on the right side of the equal sign). So 8x - 2 = 3x + 11 can be written as 8x - 3x = 11 + 2, that is, we subtracted 3x from both sides of the equal sign and added 2 to both sides of the equal sign, the resulting equation now is 5x = 13. We isolate the x by dividing both sides by 5 and our answer will be x = 13/5. This equation happens to have a unique answer, which is x = 13/5. Let us solve the equation 5x - 2 + 3x = 3(x+4) + 5x - 14. In solving this equation, we follow the same process as in steps 1 through 3 and we have the equivalent equation 8x - 2 = 8x - 2. Here, we collect our x-terms on the left side of the equal sign and our constant terms on the right side, thus giving us the equation 0x = 0 which is equal to 0=0, which is a true statement. If we look carefully at the equation, 8x - 2 = 8x - 2, we will see that for any x you substitute on both sides of the equation the results will be the same so the solution to this equation is x is real, that is, any number x will satisfy this equation. TRY IT!!! Now, let us solve the equation 5x - 2 + 3x = 3(x+4) + 5x - 10 following the same procedure as in the steps above. We will get the equation 8x - 2 = 8x + 2. We collect our x-terms on the left hand side of the equal sign and the constant terms on the right hand side of the equal sign and we will see that 0x = 4, that is, 0 = 4, not a true statement. If 0 = 4, then I could go to any bank, give them $0 and get back $4. No way. This will never happen. In this case, there is no x that will satisfy the equation given in Step #6. So the solution to this equation is: there is NO SOLUTION. Things You'll NeedAn equation of the form ax + by + c = 0 where a, b, c ∈ R, a ≠ 0 and b ≠ 0 is a linear equation in two variables. While considering the system of linear equations, we can find the number of solutions by comparing the coefficients of the equations. Also, we can find whether the system of equations has no solution or infinitely many solutions by graphical method. In this article, we will learn how to find if a system of equations has no solution or infinitely many solutions. Let us consider the pair of linear equations in two variables x and y. a1x + b1y + c1 = 0 a2x + b2y + c2 = 0 Here a1, b1, c1, a2, b2, c2 are all real numbers. Note that a12 + b12 ≠ 0, a22 + b22 ≠ 0. Case 1. If (a1/a2) = (b1/b2) = (c1/c2), then there will be infinitely many solutions. This type of equation is called a dependent pair of linear equations in two variables. If we plot the graph of this equation, the lines will coincide. Case 2. If (a1/a2) = (b1/b2) ≠ (c1/c2), then there will be no solution. This type of equation is called an inconsistent pair of linear equations. If we plot the graph, the lines will be parallel. The graph is shown below. Solved ExamplesExample 1: How many solutions does the following system have? y = -3x + 9 y = -3x – 7 (A) One solution Solution: Given equations are y = -3x + 9 y = -3x – 7 Here (a1/a2) = (b1/b2) ≠ (c1/c2). So this system of equations has no solution. Another method: Without graphing them, we can see that both have the same slope -3 which means lines are parallel. Hence the system of equations has no solution. So option (B) is the answer. Example 2: Determine whether the following system of equations have no solution, infinitely many solution or unique solutions. x+2y = 3, 2x+4y = 15 Solution: Given equations are x+2y = 3 2x+4y = 15 a1 = 1, b1 = 2, c1 = -3 a2 = 2, b2 = 4, c2 = -15 a1/a2 = ½ b1/b2 = ½ c1/c2 = 1/5 a1/a2 = b1/b2≠c1/c2 So, the system of equations has no solution. Related Links:
Frequently Asked QuestionsA Linear equation is an equation that has one or more variables having degree one. An example of a Linear equation in two variables: 2x + 3y + 4 = 0. If a1/a2 = (b1/b2) ≠
(c1/c2), then there will be no solution. If (a1/a2) = (b1/b2) = (c1/c2), then there will be
infinitely many solutions. |
How do you know when an equation has no solution
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