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Show Systems of Linear equations:A system of linear equations is just a set of two or more linear equations. In two variables ( x and y ) , the graph of a system of two equations is a pair of lines in the plane. There are three possibilities:
How to Solve a System Using The Substitution Method
Note 1 : If it's easier, you can start by solving an equation for x in terms of y , also – same difference! Example: Solve the system { 3 x + 2 y = 16 7 x + y = 19 Solve the second equation for y . y = 19 − 7 x Substitute 19 − 7 x for y in the first equation and solve for x . 3 x + 2 ( 19 − 7 x ) = 16 3 x + 38 − 14 x = 16 − 11 x = − 22 x = 2 Substitute 2 for x in y = 19 − 7 x and solve for y . y = 19 − 7 ( 2 ) y = 5 The solution is ( 2 , 5 ) .Note 2 : If the lines are parallel, your x -terms will cancel in step 2 , and you will get an impossible equation, something like 0 = 3 . Note 3 : If the two equations represent the same line, everything will cancel in step 2 , and you will get a redundant equation, 0 = 0 . In our last lesson, we introduced the topic of solving linear systems in two variables. We learned how to solve a linear system using graphing. Although we can obtain a solution using graphing, the method is not very practical. In this lesson, we will focus on another method to solve a linear system known as "substitution". The substitution method is most useful when one of the coefficients for one of the variables is either 1 or -1. Solving a Linear System using the Substitution Method
Let's look at a few examples. Special Case Linear SystemsIf both variables drop out when solving a system of linear equations:
Linear Systems with No Solution In some cases, we will not have a solution for our linear system. This will occur when we have two parallel lines. This type of system is known as an "inconsistent system". If we are solving our system using the substitution method, we will notice that our variables disappear and we are left with a false statement. Let's look at an example. Linear Systems with Infinitely Many Solutions Another special case scenario occurs when the same equation is presented twice as a system of
equations. In this case, what works as a solution for one equation works as a solution to the other. These equations are known as "dependent equations". Let's look at an example. Skills Check:Example #1 Solve each system using substitution. $$-46x - 19y=-38$$ $$92x + y=2$$ Please choose the best answer. Example #2 Solve each system using substitution. $$6x - 8y=0$$ $$-11x + 15y=19$$ Please choose the best answer. Example #3 Solve each system using substitution. $$27x + 7y=147$$ $$x + y=1$$ Please choose the best answer. Congrats, Your Score is 100% Better Luck Next Time, Your Score is % Try again?
How do you solve an equation with two substitutions?To solve systems using substitution, follow this procedure:. Select one equation and solve it for one of its variables.. In the other equation, substitute for the variable just solved.. Solve the new equation.. Substitute the value found into any equation involving both variables and solve for the other variable.. How do you do substitution method?Substitution Method Steps. Simplify the given equation by expanding the parenthesis.. Solve one of the equations for either x or y.. Substitute the step 2 solution in the other equation.. Now solve the new equation obtained using elementary arithmetic operations.. |