Problem 15 - College Algebra

Question

Use Cramer's Rule to solve each system. $$\left\\{\begin{array}{l}4 x-5 y=17 \\\2 x+3 y=3\end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system of equations is x = 3 and y = -1.

Step 1 - Setup of the Main Determinant

First, setup the main determinant that holds all coefficients found in the systems of equation. This determinant is defined as: \[ D = \left| \begin{array}{cc} 4 & -5 \ 2 & 3 \ \end{array} \right| \]

Step 2 - Calculation of Determinant

The determinant of D is calculated using the formula \(ad-bc\). Hence\[ D = 4 * 3 - (-5) * 2 = 12 + 10 = 22 \]

Step 3 - Calculation of D_x

Next, calculate the determinant for x, D_x, by replacing the coefficients of x with the constant term on the right side. Hence\[ D_x = \left| \begin{array}{cc} 17 & -5 \ 3 & 3 \ \end{array} \right| \]And find the value of D_x by using the formula \(ad-bc\).\[ D_x = 17*3 - (-5)*3 = 51 + 15 = 66 \]

Step 4 - Calculation of D_y

Calculate the determinant for y, D_y, by replacing the coefficients of y with the constant term on the right side. Hence \[ D_y = \left| \begin{array}{cc} 4 & 17 \ 2 & 3 \ \end{array} \right| \]And find the value of D_y by using the formula \(ad-bc\).\[ D_y = 4*3 - 17*2 = 12 - 34 = -22 \]

Step 5 - Calculation of Result

After finding determinants D, D_x and D_y, we can now calculate the values of x and y using the formulas: \(x = D_x / D\) and \(y = D_y / D\). Thus:\[ x = D_x / D = 66 / 22 = 3 \]\[ y = D_y / D = -22 / 22 = -1 \]

Key Concepts

Understanding Systems of Linear Equations

Linear equations are like recipes that tell us how to mix certain ingredients to get a desired result. But rather than mixing in the kitchen, we're mixing variables like x and y to get specific solutions.

A system of linear equations is simply a set of two or more equations with the same variables. For example our given exercise contains a pair of equations:

• 4x - 5y = 17
• 2x + 3y = 3

Imagine x and y as ingredients and the numbers 17 and 3 as the results of the recipes. The goal? To find out just the right amount of x (let's say chocolate) and y (maybe butter) needed to achieve those perfect results (delicious cakes). When we say solve the system, we mean finding the exact values of x and y that make both equations true. It's like discovering the secret recipe!

Determinant Calculation Made Easy

Now, how do we unravel this secret recipe? This is where determinants are the ace up our sleeves.

Imagine determinants as a special tool that helps us quickly evaluate the mixing ability of our ingredients in the context of our equations. They're like a blender's settings that tell us if we can blend our ingredients to match the desired cake or not. In our problem, we calculate a determinant by taking a square array of numbers (our coefficients) and doing some cross-multiplication and subtraction.

For instance, our main determinant D looks like this:

• D = | 4 -5 |
• | 2 3 |

To calculate D, imagine drawing lines from the top left number to the bottom right (4 to 3) and then from top right to bottom left (-5 to 2). Multiply the numbers along each line and then subtract one result from the other. Voila, you've got your determinant!

Algebraic Methods and Cramer's Rule

Algebraic methods are your toolkit for solving equations. Just like a mechanic uses a wrench to fix a car, algebraic methods help 'fix' equations to get the solutions we need. Cramer's Rule is one special tool in our algebra toolkit. Known for its precision, it's like an electric screwdriver, making our work quicker and more efficient.

Cramer's Rule uses the concept of determinants to solve systems of linear equations. It tells us that we can find the value of each variable by taking the determinant of a modified version of the main matrix, then dividing by the main determinant. It's like knowing you can swap out ingredients in a recipe and still get a perfect cake by adjusting the portions. Using Cramer's Rule, we tweak our matrix to focus on one variable at a time, replace some numbers, and follow a similar calculation as before. Before we know it, the values of x and y are served!