Question

Use linear combinations to solve the system. $$ \begin{aligned} &-x+2 y=12\\\ &x+6 y=20 \end{aligned} $$

Short answer

The solution to the system of equations is \(x = 8\) and \(y = 2\).

Step-by-step solution

Arranging the equations

Rearrange the two equations as: \[\begin{align*}x &= 20 - 6y \x &= 12 - 2y\end{align*}\]This is done because the method of linear combinations involves adding or subtracting equations to eliminate one variable, which makes it easier to isolate one equation in either variable.

Combine the equations

Set the two equations equal to each other to eliminate x:\[20 - 6y = 12 - 2y\]This creates a single equation with only one variable, y.

Solve for y

Solving the single-variable equation, bring like terms together:\[-6y + 2y = 12 - 20\]which simplifies to:\[-4y = -8\]Dividing both sides by -4 gives \[y = 2\]

Solve for x

Substitute the obtained value of y into either of the original equations to solve for x. Let us substitute y = 2 into the second equation:\[x = 20 - 6*2 \ x = 20 - 12 = 8\]

Key concepts

Linear Combinations Method

The linear combinations method, also known as the addition or elimination method, is a powerful technique in algebra for solving systems of linear equations. This method involves combining the given equations in such a way as to eliminate one of the variables, allowing you to easily solve for the other.

The process usually involves multiplying each equation by a suitable number so that when the equations are added or subtracted from each other, one of the variables is canceled out. The steps you take will vary depending on the system, but the goal is always to reduce the complexity and solve for variables one at a time with simpler, one-variable equations.

Systems of Equations

A system of equations is a set of two or more equations that have the same variables. The solution to a system of equations is the set of variable values that make all of the equations true simultaneously. When dealing with two variables, the graphical interpretation would be finding the point of intersection of the lines represented by the equations.

Systems can have one solution (intersecting lines), no solution (parallel lines), or infinitely many solutions (coincident lines). Understanding the nature of the lines and the system is crucial in choosing the appropriate method for finding a solution.

Algebraic Manipulation

Algebraic manipulation is the process of using algebraic operations to simplify expressions or solve equations and inequalities. This can involve a variety of techniques, such as expanding brackets, factorizing, simplifying fractions, and combining like terms.

Effective manipulation is essential for rearranging equations into a form that is easier to work with. For example, in solving systems of equations, you must often rearrange the equations to get the coefficients of a variable to match so that the elimination method can be used. Developing strong skills in algebraic manipulation can make the difference between finding a solution easily or struggling with the problem.

Substitution Method

The substitution method is another technique for solving systems of equations. This method involves solving for one variable in terms of the other and then substituting this expression into the other equation. This effectively reduces a system of two equations to a single equation with one variable, which can then be solved.

Substitution is particularly useful for systems where one of the variables is already isolated or can be easily isolated. Once a value is found for one variable, it's substitution back into one of the original equations to solve for the remaining variable completes the process.