Solving 2x2 Systems of Equations by Substitution
Description/Explanation/Highlights
Video Description
This video explains how to solve a 2×2 system of equations by substitution.
Steps and Key Points to Remember
To solve a 2×2 system of equations by substitution, follow these steps:
- To solve a system of equations by substitution, pick one of the equations and solve for either x or y.
- Substitute the solution into the x or y of the other equation and solve.
- Plug that solution into one of the equations…usually the one you solved for x or y is easier…to solve for the other variable.
- In the example system, \(x-y=-11\) and \(7x+4y=-22\), pick one of the equations and solve for one of the variables. The first equation looks easier because there are no coefficients in front of x or y and x looks easier to solve for since there is no negative sign in front of it. It would work for any variable on either equation though.
- Adding y to both sides of the equation results in \(x=y-11\) and we have solved for x.
- Substitute the value for x, \(y-11\), into the x in the other equation. \(7(y-11)+4y=-22\)
- Solve for y by distributing, getting y to one side and numbers to the other, and dividing.
\(7(y-11)+4y=-22\)
\(7y-77+4y=-22\) <==distribute
\(11y=55\) <==combine terms and get variables on one side of the = and numbers on the other
\(y=5\) <==divide to solve
\(7y-77+4y=-22\) <==distribute
\(11y=55\) <==combine terms and get variables on one side of the = and numbers on the other
\(y=5\) <==divide to solve
- Now substitute 5 in for y in the equation you found for x above: \(x=(5)-11\) and \(x=-6\).
- The solutions are \(x=-6\) and \(y=5\) or \((-6,5)\)
- Check by substituting -6 in for x and 5 in for y in both equations to be sure they create true statements.
- When deciding which equation to solve and which variable to solve for, look for small coefficients, numbers that will divide with whole number answers, and positive numbers to make solving easier.
- For example, in the system: \(3x+9y=51\) and \(2x+4y=18\), it may be easier to solve the second equation since all the coefficients divide by 2 (one of the coefficients) and the numbers are smaller. Also since 2 is the coefficient of x, solving for x will probably be easier. You could choose either equation or either variable but doing so could create fractions and harder to solve literal equations.
- If substituting causes a false statement, the system has no solution.
- For example in the system: \(5x+y=2\) and \(10x+2y=9\), since y has a coefficient of 1 in the first equation, we will solve for y by moving 5x to the right side of the equal sign and changing the sign resulting in \(y=-5x+2\).
- Substituting in for y in the second and solving creates a false statement. \(10x+2(-5x+2)=9\longrightarrow 10x-10x+4=9\longrightarrow 4\neq9\), therefore there is no solution.
- If solving results in a true statement with no variable such as \(9=9\), there are infinite solutions to the system of equations.
Here are some key points to keep in mind when solving a 2×2 system of equations by substitution.
- Pick one equation and one variable to solve for in that equation.
- Substitute the solution into the other equation and solve.
- It will be easier to solve if you choose the equation you will solve and the variable you solve carefully.
- Choose equations whose coefficients are one, have small numbers, divide to create whole numbers, and are positive to make solving easier.
- If there are no good choices for equations and variables to solve, consider other methods of solving the system.
- Check the solution by substituting in the solution you get into x and y in both equations and making sure true statements are created.
Video Highlights
- 00:00 Introduction
- 00:16 Solve the system, \(x-7=-11\) and \(7x+4y=-22\) by substitution.
- 02:38 \(3x+9y=51\) and \(2x+4y=18\) example of solving and checking using substitution
- 05:00 \(5x+y=2\) and \(10x+2y=9\) example of solving a system with no solution using substitution
- 07:00 Conclusion
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