Problem 9 - Elementary Algebra Within Reach

Question

In Exercises 7-12, solve the system by the method of elimination. $$ \left\\{\begin{array}{r} -x+2 y=6 \\ 2 x+5 y=6 \end{array}\right. $$

Step-by-Step Solution

Answer

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

Step 1: Multiply the Equations

To prepare for elimination, we'll need to ensure that adding the two equations will cancel out one variable. If we multiply the first equation by 2 and the second equation by 1, we will get the following: \n\n\(2(-x+2y) = 2*6 \) \ \n\(1*(2x+5y) = 1*6 \) \ \nThis simplifies to: \n\n\[-2x+4y =12 \] \n\n\[2x+5y = 6 \]

Step 2: Add the Adjusted Equations and Solve for One Variable

Let's now add the adjusted equations to eliminate \(x\): \[-2x + 2x + 4y + 5y = 12 + 6 \rightarrow 9y = 18. \] Dividing each side by 9 gives \(y=2\).

Step 3: Substitute and Solve for Remaining Variable

We will substitute \(y\) from step 2 into the original second equation to solve for \(x\): \ \[2x+5*2= 6 \rightarrow 2x+10 = 6 \rightarrow 2x =6-10 = -4. \] Dividing by 2, we require \(x=-2\).

Key Concepts

Elimination Method

The elimination method is a strategic approach to solving systems of equations, where you aim to eliminate one variable so you can easily solve for another. This method involves aligning two equations so that when they are added or subtracted, one variable cancels out due to its coefficients being opposites.In our textbook problem, to prepare for elimination, the first equation was multiplied by 2 to set up the coefficients of variable x to be opposites. This smart adjustment brought us to a point where adding both equations resulted in eliminating x and leaving an equation with just one variable, y, making it easier to solve.

Systems of Equations

A system of equations is a set of two or more equations with the same variables. Our goal with such a system is to find the values of those variables that satisfy all equations simultaneously.

There are several methods for solving such systems, and the elimination method is just one of those. It's noteworthy that different systems might lend themselves more readily to different methods of solution, and in this case, elimination was the chosen method due to the easily manipulated coefficients.

Algebraic Operations

Algebraic operations, which include addition, subtraction, multiplication, and division, are the bread and butter of algebra. They're used to manipulate algebraic expressions and equations to simplify them or solve for unknowns.

These operations follow specific properties and rules to maintain the equations' balance. For instance, what's done to one side of the equation must be done to the other. In the provided solution, multiplication was first used to align the coefficients, and then addition was employed to eliminate a variable.

Substitution

Once we have the value of one variable, we employ substitution to find the other. Substitution involves taking the known value of a variable and plugging it into an equation where that variable and another are present.

This step turns an equation with two variables into an equation with just one, which can then be solved using basic algebraic operations. In our textbook problem, after finding the value of y, it was substituted back into one of the original equations to find the value of x.