Problem 26 - Algebra and Trigonometry

Question

Solve each system by the addition method. \(\left\\{\begin{array}{l}{3 x-7 y=13} \\ {6 x+5 y=7}\end{array}\right.\)

Step-by-Step Solution

Answer

The solutions for the system of equations are \(x=1, y=2\) and \(x=-1, y=2\).

Step 1: Rearranging the second equation

Rearrange the second equation by performing arithmetic operation as follows: \((x^{2}+(y-8)^{2}=41)\) leading to \(x^{2}+y^{2}-16y+64=41\) and finally to \(x^{2}+y^{2}-16y= -23\).

Step 2: Subtract the rearranged second equation from the first one

Perform the subtraction \(y^{2}- (-16y) = 5 -(-23)\) and simplify to get \(y=2\).

Step 3: Substitute y into one of the original equations

We use the first equation \(x^{2}+y^{2}=5\). Putting \(y=2\) into this equation gives \(x^{2}+2^{2}=5\), simplifying leads to \(x^{2}=1\).

Step 4: Solve for x

\(\sqrt{x^{2}}=\sqrt{1}\) gives \(x=1\) and \(x=-1\) as solutions.

Key Concepts

Addition Method

The addition method is a handy tool when solving systems of equations. It involves aligning two equations and then adding or subtracting them to eliminate one variable. The main goal is to simplify the system of equations, making it easier to find the values of the unknowns.

In the given exercise, we have two equations. By rearranging the second equation, we can make the same terms in the equations align properly. This reorganization is crucial because it sets the stage for effectively using addition or subtraction on the equations. The beauty of this method lies in its simplicity and the straightforward arithmetic involved.

Quadratic Equations

Quadratic equations appear frequently in algebra, typically in the form of \[ax^2 + bx + c = 0\]. In the exercise presented, the appearance of terms such as \(x^2\) and \(y^2\) indicates we are dealing with quadratic equations, even though it's part of a system.

When handling quadratic equations, especially in systems, understanding their structure helps simplify the process. Recognizing terms like \(x^2\) is important because you will need to handle their special characteristics, such as having up to two real solutions.

Algebraic Manipulation

Algebraic manipulation is the art of rearranging and simplifying expressions to reveal the desired solutions. It often involves operations such as addition, subtraction, multiplication, division, and sometimes factoring.

In the solution steps, the second equation was rearranged to match terms with the first equation, enabling the addition method to work. Specifically, the expression \((y-8)^2\) was expanded to \(y^2 - 16y + 64\), and rearranged to combine like terms on one side. This manipulation simplifies problems into more manageable pieces, highlighting the power of methodical arithmetic operations to solve algebra problems efficiently.

Substitution Method

The substitution method involves solving one of the equations for one variable and then substituting that expression into the other equation. This approach isolates one variable, making it easier to solve the remaining equation.

In the provided solution, once we find \(y=2\), substituting this value into one of the original equations \(x^2 + y^2 = 5\) simplifies our work by reducing the number of variables. This substitution gives a straightforward quadratic equation in one variable, \(x^2 = 1\), which is easy to solve. The substitution method is particularly useful when one of the equations in the system is linear, but it can also be applied to systems that include quadric forms.