Question

Solve by determinants. $$\begin{aligned} &\frac{3 x}{5}+\frac{2 y}{3}=17\\\ &\frac{2 x}{3}+\frac{3 y}{4}=19 \end{aligned}$$

Short answer

The solution to the system is \(x = \frac{(255)(9) - (10)(228)}{(9)(9) - (10)(8)}\) and \(y = \frac{(9)(228) - (255)(8)}{(9)(9) - (10)(8)}\).

Step-by-step solution

Write the system of equations in standard form

Multiply each equation by the least common multiple (LCM) of the denominators to eliminate the fractions and write the system in standard form. For the first equation, the LCM of 5 and 3 is 15, so multiply both sides by 15 to get the first equation in standard form: \((3x/5) * 15 + (2y/3) * 15 = 17 * 15\). Do the same for the second equation by finding the LCM of 3 and 4, which is 12, and then multiplying both sides by 12.

Calculate the first equation

After multiplying, simplify the first equation to get rid of the denominators: \(9x + 10y = 255\).

Calculate the second equation

Similarly, for the second equation, after multiplying and simplifying, you get: \(8x + 9y = 228\).

Set up the determinant equations

Use Cramer's Rule to solve the system by setting up two determinants for x and y. The denominator determinant, D, uses the coefficients of x and y: \(D = |\begin{array}{cc}9 & 10 \ 8 & 9\end{array}|\). The numerator determinant for x, Dx, replaces the x-coefficients with the constants: \(Dx = |\begin{array}{cc}255 & 10 \ 228 & 9\end{array}|\), and the numerator determinant for y, Dy, replaces the y-coefficients: \(Dy = |\begin{array}{cc}9 & 255 \ 8 & 228\end{array}|\).

Calculate determinants

Compute D, Dx, and Dy using the formula for the determinant of a 2x2 matrix, \(|\begin{array}{cc}a & b \ c & d\end{array}| = ad - bc\). Thus, \(D = (9)(9) - (10)(8)\), \(Dx = (255)(9) - (10)(228)\), and \(Dy = (9)(228) - (255)(8)\).

Solve for x and y

Divide the numerators by the denominator to find the values of x and y using Cramer's Rule: \(x = Dx/D\) and \(y = Dy/D\).

Key concepts

Cramer's Rule

Cramer's Rule is a straightforward method used in solving systems of linear equations with as many equations as variables. It uses determinants to find the solution for each variable. For a system of two equations, the rule defines a separate determinant for each variable, where the determinant called the numerator replaces the corresponding column of coefficients with the constants from the right-hand side of the equations.

However, the denominator determinant remains the same for all variables and uses only the coefficients from the left-hand side of the system. Once you calculate all necessary determinants, you solve for each variable by dividing its numerator determinant by the denominator. It's important to remember that Cramer's Rule only works when the denominator determinant is non-zero.

Let's look into our example from the exercise. After eliminating fractions and simplifying, we have two equations in the standard form: \(9x + 10y = 255\) and \(8x + 9y = 228\). Using Cramer's Rule, we set up separate determinants for \(x\) and \(y\), which are referred to as \(D_x\) and \(D_y\). These are then divided by the common determinant, \(D\), to find the values of \(x\) and \(y\).

Determinants in Linear Algebra

Determinants play a critical role in linear algebra, especially when solving systems of equations. A determinant is a special number that can be calculated from a square matrix. In the context of a system of linear equations, the determinant helps us to understand whether the system has a unique solution, no solution, or infinitely many solutions.

If the determinant of the coefficient matrix (the matrix formed by the coefficients of the variables) is zero, the system does not have a unique solution. Conversely, if the determinant is non-zero, the system has a unique solution which can be computed using techniques such as Cramer's Rule mentioned previously.

In our exercise with two variables, we calculate three determinants: \(D\), the denominator determinant using the coefficents of the variables, and \(D_x\) and \(D_y\), the numerator determinants for variables \(x\) and \(y\). The process for calculating a 2x2 determinant, like the ones we use in the exercise, is to multiply the top-left and bottom-right entries and subtract the product of the top-right and bottom-left entries. These values directly impact the final solution when applying Cramer's Rule.

Elimination of Fractions in Equations

When solving equations, fractions can often complicate matters. To simplify the process, we aim to eliminate fractions before proceeding to solve the system of equations. This is typically achieved by multiplying each term of the equation by the least common multiple (LCM) of the denominators of the fractions present in the equation. Doing so will clear the fractions and convert the equation into a form that is more manageable to work with.

In the exercise we're discussing, the system of equations initially contains fractions, making them harder to handle. By determining the LCM of the denominators in each equation and subsequently multiplying every term by these LCMs, we successfully remove the fractions and bring each equation into standard linear form.

It's essential when removing fractions to be meticulous with arithmetic to prevent any mistakes that could lead to incorrect solutions. Once we have our equations free of fractions, we can apply methods like Cramer's Rule to find our variables' values.

Standard Form Linear Equations

A standard form linear equation is typically represented as \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers, and \(A\) and \(B\) are not both zero. This form is particularly useful as it facilitates the application of various algebraic methods, and is a prerequisite for applying Cramer's Rule.

The first step in solving our exercise problem is to transform the given equations with fractions into standard form by eliminating the denominators. Once in standard form, the coefficients of \(x\) and \(y\) neatly fall into place in the matrix form, which is essential for finding the determinants needed for Cramer's Rule.

It's also worth mentioning that when equations are in standard form, they provide a clear and uniform structure making systems easier to compare and contrast with one another. Additionally, graphing these equations becomes more straightforward since the standard form readily yields the intercepts.