Problem 99 - Algebra and Trigonometry

Question

Solve the system for \(x\) and \(y\) in terms of \(a_{1}, b_{1}, c_{1}, a_{2}, b_{2},\) and \(c_{2}\) $$\left\\{\begin{array}{l} {a_{1} x+b_{1} y=c_{1}} \\ {a_{2} x+b_{2} y=c_{2}} \end{array}\right.$$

Step-by-Step Solution

Answer

The solutions for \(x\) and \(y\) are \n \(x = (c_{1} a_{1} - b_{1} (a_{1} c_{2} - a_{2} c_{1})) / (a_{1} (a_{2} b_{1} - a_{1} b_{2}))\) and \n \(y = (a_{1} c_{2} - a_{2} c_{1}) / (a_{2} b_{1} - a_{1} b_{2})\) respectively.

Step 1: Solve the first equation for x

In order to use the substitution method, we first need to solve one equation for one of the variables. In this case we will start with solving the first equation \(a_{1} x + b_{1} y = c_{1}\) for variable \(x\).\n This gives: \n \(x = (c_{1} - b_{1} y) / a_{1}\)

Step 2: Substitute into the second equation

Now substitute \(x\) into the second equation: \n \(a_{2} ((c_{1} - b_{1} y) / a_{1}) + b_{2}y = c_{2}\). Multiply through by \(a_{1}\) to easily solve for \(y\): \n \( a_{2} c_{1} - a_{2} b_{1} y + a_{1} b_{2} y = a_{1} c_{2}\).

Step 3: Solve for y

Rearrange this equation to solve for \(y\): \n \( (a_{2} b_{1} - a_{1} b_{2}) y = a_{1} c_{2} - a_{2} c_{1}\). Then, solve for \(y\): \n \(y = (a_{1} c_{2} - a_{2} c_{1}) / (a_{2} b_{1} - a_{1} b_{2})\).

Step 4: Substitute \(y\) back into \(x\)

Now, substitute the value of \(y\) back into \(x = (c_{1} - b_{1} y) / a_{1}\). Therefore, \n \(x = (c_{1} a_{1} - b_{1} (a_{1} c_{2} - a_{2} c_{1})) / (a_{1} (a_{2} b_{1} - a_{1} b_{2}))\)

Key Concepts

Substitution Method

The substitution method is one of the strategies to solve a system of equations. It involves solving one of the equations for one variable and then substituting this expression into the other equation. This helps to simplify the system by eliminating one variable, making it easier to solve for the other.

To apply this method:

• Choose one equation and solve for one variable in terms of the other. It’s usually more convenient to choose an equation and a variable that will be easy to isolate.
• Substitute the expression you found into the other equation. This effective step will yield an equation with only one variable.
• Solve this new equation for the remaining variable.

Once you have solved for one variable, substitute this value back into the expression for the second variable to find its value as well.

This method is highly systematic and works well, especially when one of the variables can be easily isolated in one of the equations.

System of Equations

A system of equations is a set of two or more equations with the same set of unknowns or variables. Understanding systems of equations is fundamental in algebra as they represent real-world situations where multiple conditions or constraints exist. These equations can be solved using various methods like substitution, elimination, or graphing, each having its own advantages.

In our exercise, we are given a system of two linear equations:\[\left\{\begin{array}{l}{a_{1} x+b_{1} y=c_{1} }\ {a_{2} x+b_{2} y=c_{2}}\end{array}\right.\]Here, both equations share the variables \(x\) and \(y\), and the goal is to find the values of these variables that satisfy both equations simultaneously. When dealing with linear equations like these, the system can produce one unique solution, infinitely many solutions, or no solution at all, depending on the nature of the lines represented by the equations.

Variables

Variables are symbols, often letters, used to represent unknown or changeable values in mathematical equations. They are fundamental components that allow the formulation of relationships between different quantities.

In our system of equations:

• The letters \(x\) and \(y\) are the variables for which we are solving.
• Constants like \(a_{1}, b_{1}, c_{1}, a_{2}, b_{2},\) and \(c_{2}\) are given values that multiply or add with these variables, defining the specific conditions of the equations.

Understanding variables and their roles in equations allows us to manipulate and solve equations systematically. They provide the means to create general formulas and models that describe real and theoretical situations.

Solving for y

After using the substitution method, we often encounter an equation that contains a single variable, which we then need to solve. Solving for a particular variable means rearranging an equation so that this variable is on one side by itself.

In our example, once we substitute for \(x\), we get a simplified equation involving only \(y\):\[(a_{2} b_{1} - a_{1} b_{2}) y = a_{1} c_{2} - a_{2} c_{1}\]To solve for \(y\), follow these steps:

• Isolate \(y\) by dividing both sides of the equation by the coefficient of \(y\). In this case:\[y = \frac{a_{1} c_{2} - a_{2} c_{1}}{a_{2} b_{1} - a_{1} b_{2}}\]
• This result gives us the value of \(y\) in terms of the other constants present in the system.

Once \(y\) is found, it can be substituted back into the equation for \(x\), thus solving the entire system of equations.