8 3 Systems Of Linear Equations Solving By Substitution

8 3 Systems Of Linear Equations Solving By Substitution Cracking the Code Solving Systems of Linear Equations by Substitution Solving systems of linear equations is a fundamental skill in algebra with applications ranging from basic problemsolving to complex modeling in various fields One powerful technique for tackling these systems is the substitution method This method involves expressing one variable in terms of the other and then substituting this expression into the remaining equation to solve for a single variable Heres a breakdown of the substitution method illustrated with eight diverse examples Understanding the Basics Before diving into the examples lets quickly recap the core concepts System of Linear Equations A set of two or more linear equations with the same variables Solution A set of values for the variables that satisfy all equations in the system Substitution Method A technique to solve systems of equations by isolating one variable in one equation and substituting it into the other equation Example 1 Simple System Equations x y 5 x y 1 Steps 1 Isolate a variable Lets isolate x from the second equation x y 1 2 Substitute Substitute this expression for x in the first equation y 1 y 5 3 Solve for y Simplify and solve 2y 1 5 2y 4 y 2 4 Solve for x Substitute the value of y 2 back into either original equation Lets use the first equation x 2 5 x 3 Solution The solution is x 3 y 2 Example 2 Isolating a Variable with a Coefficient Equations 2 2x 3y 11 x y 2 Steps 1 Isolate x From the second equation x y 2 2 Substitute Substitute this expression for x in the first equation 2y 2 3y 11 3 Solve for y Simplify and solve 2y 4 3y 11 5y 7 y 75 4 Solve for x Substitute y 75 back into either original equation Lets use the second equation x 75 2 x 175 Solution The solution is x 175 y 75 Example 3 System with Fractional Coefficients Equations 12x 13y 1 14x 16y 0 Steps 1 Isolate x From the second equation 14x 16y x 23y 2 Substitute Substitute this expression for x in the first equation 1223y 13y 1 3 Solve for y Simplify and solve 13y 13y 1 23y 1 y 32 4 Solve for x Substitute y 32 back into either original equation Lets use the second equation 14x 1632 0 14x 14 x 1 Solution The solution is x 1 y 32 Example 4 System with Negative Coefficients Equations 3x 2y 4 x y 2 Steps 1 Isolate x From the second equation x y 2 2 Substitute Substitute this expression for x in the first equation 3y 2 2y 4 3 Solve for y Simplify and solve 3y 6 2y 4 y 2 y 2 4 Solve for x Substitute y 2 back into either original equation Lets use the second equation x 2 2 x 0 Solution The solution is x 0 y 2 Example 5 System with Variables on Both Sides Equations 2x 5y 1 3 3x 2y 11 Steps 1 Isolate x From the second equation 3x 2y 11 x 23y 113 2 Substitute Substitute this expression for x in the first equation 223y 113 5y 1 3 Solve for y Simplify and solve 43y 223 5y 1 193y 193 y 1 4 Solve for x Substitute y 1 back into either original equation Lets use the second equation 3x 21 11 3x 9 x 3 Solution The solution is x 3 y 1 Example 6 System with Two Variables and Two Constants Equations 4x 3y 10 2x 5y 4 Steps 1 Isolate x From the second equation 2x 5y 4 x 52y 2 2 Substitute Substitute this expression for x in the first equation 452y 2 3y 10 3 Solve for y Simplify and solve 10y 8 3y 10 13y 18 y 1813 4 Solve for x Substitute y 1813 back into either original equation Lets use the second equation 2x 51813 4 2x 4613 x 2313 Solution The solution is x 2313 y 1813 Example 7 System with Fractional Coefficients and Variables on Both Sides Equations 12x 23y 5 34x 16y 1 Steps 1 Isolate x From the second equation 34x 16y 1 x 29y 43 2 Substitute Substitute this expression for x in the first equation 1229y 43 23y 5 3 Solve for y Simplify and solve 19y 23 23y 5 79y 133 y 397 4 Solve for x Substitute y 397 back into either original equation Lets use the second equation 34x 16397 1 34x 8342 x 8363 Solution The solution is x 8363 y 397 Example 8 System with Complex Expressions Equations 4 2x y 3x 4 3x y 5y 7 Steps 1 Simplify the equations 2x 2y 3x 4 and 3x 3y 5y 7 2 Isolate x From the first simplified equation 2y x 4 x 2y 4 3 Substitute Substitute this expression for x in the second simplified equation 32y 4 3y 5y 7 4 Solve for y Simplify and solve 6y 12 3y 5y 7 2y 5 y 52 5 Solve for x Substitute y 52 back into either simplified equation Lets use the first simplified equation 2x 252 3x 4 x 9 Solution The solution is x 9 y 52 Key Points to Remember Choose the Easiest Variable When selecting a variable to isolate aim for the one with the simplest coefficient ideally 1 or 1 Be Careful with Signs Pay close attention to the signs when substituting and solving for the variables Check Your Solution After finding your solution substitute the values back into both original equations to verify that they are satisfied Conclusion The substitution method provides a systematic approach to solving systems of linear equations By isolating one variable and substituting its expression into the other equation you can reduce the problem to a single variable equation making it easier to solve Whether you are working with simple or complex systems this method offers a reliable and efficient way to find the solution