Solving Systems Of Equations Is The Expression For X Correct

Hey math enthusiasts! Today, we're diving deep into the world of linear equations, specifically a system of two equations with two unknowns. We've got a juicy problem on our hands, and we're going to dissect it piece by piece to determine if a given expression truly represents the value of x in the solution. So, buckle up, grab your pencils, and let's get started!

The Challenge: Unraveling the Equations

Our mission, should we choose to accept it (and we do!), is to analyze the following system of equations:

$$-2x + 3y = -3$$

$$3x - 4y = 5$$

And the question that burns in our minds is this: Does the following expression accurately give us the value of x?

$$x = \frac{\begin{bmatrix} -3 & 3 \\ 5 & -4 \end{bmatrix}}{\begin{bmatrix} -2 & 3 \\ 3 & -4 \end{bmatrix}}$$

This looks like a job for determinants, those magical numbers that pop out of matrices and help us solve systems of equations. But before we jump into calculations, let's make sure we understand the underlying principle: Cramer's Rule. This is the key to unlocking the mystery of this problem.

Cramer's Rule: Our Guiding Light

Cramer's Rule is a powerful tool for solving systems of linear equations using determinants. It states that for a system of equations like this:

$$ax + by = e$$

$$cx + dy = f$$

The solution for x and y can be found using the following formulas:

$$x = \frac{\begin{vmatrix} e & b \\ f & d \end{vmatrix}}{\begin{vmatrix} a & b \\ c & d \end{vmatrix}}$$

$$y = \frac{\begin{vmatrix} a & e \\ c & f \end{vmatrix}}{\begin{vmatrix} a & b \\ c & d \end{vmatrix}}$$

Notice the pattern here, guys? The denominator is the same for both x and y. It's the determinant of the coefficient matrix, which is formed by the coefficients of x and y in our equations. For x, the numerator's determinant is formed by replacing the x-coefficient column in the coefficient matrix with the constants on the right side of the equations. A similar thing happens for y, but we replace the y-coefficient column instead. This rule provides a structured way to calculate x and y, making it especially useful for larger systems of equations.

Applying Cramer's Rule to Our Problem

Now, let’s apply Cramer's Rule to our specific system. First, let's identify our a, b, c, d, e, and f:

• a = -2
• b = 3
• c = 3
• d = -4
• e = -3
• f = 5

Now, let's plug these values into Cramer's Rule for x:

$$x = \frac{\begin{vmatrix} -3 & 3 \\ 5 & -4 \end{vmatrix}}{\begin{vmatrix} -2 & 3 \\ 3 & -4 \end{vmatrix}}$$

Hold on a second! This looks exactly like the expression we were given in the problem. But we're not done yet. We need to actually calculate the determinants to find the value of x and confirm whether the expression is truly correct. Understanding Cramer's Rule is only half the battle; we need to wield its power to get to the solution.

Calculating the Determinants: Time for Some Math!

The determinant of a 2x2 matrix $\begin{bmatrix} a & b \ c & d \end{bmatrix}$ is calculated as (ad - bc). So, let's calculate the determinant in the numerator:

Determinant (Numerator) = (-3 * -4) - (3 * 5) = 12 - 15 = -3

Now, let's calculate the determinant in the denominator:

Determinant (Denominator) = (-2 * -4) - (3 * 3) = 8 - 9 = -1

Therefore, the value of x is:

$$x = \frac{-3}{-1} = 3$$

The Verdict: True or False?

So, the expression provided does indeed represent the setup for finding the value of x using Cramer's Rule. We've gone through the process step-by-step, from understanding the rule itself to applying it to our specific system of equations and finally calculating the determinants. However, the expression itself only represents the setup; we needed to calculate the determinants to find the actual value of x, which is 3. This meticulous approach highlights the importance of not just recognizing the formula but also executing the calculations to arrive at the final answer. Our journey through this problem showcases the core of mathematical problem-solving: understanding, applying, and verifying.

Exploring Alternative Solutions: Beyond Cramer's Rule

While Cramer's Rule is a fantastic tool, it's not the only way to solve a system of linear equations. Let's briefly explore two other methods: substitution and elimination. These alternative approaches provide different perspectives on solving systems of equations and can be valuable additions to your mathematical toolkit.

Substitution Method: A Swapping Strategy

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This effectively reduces the system to a single equation with one variable, which we can then solve. Let's see how it works with our example:

1. Solve the first equation for y:

   $$-2x + 3y = -3$$

   $$3y = 2x - 3$$

   $$y = \frac{2x - 3}{3}$$

2. Substitute this expression for y into the second equation:

   $$3x - 4(\frac{2x - 3}{3}) = 5$$

3. Simplify and solve for x:

   $$9x - 8x + 12 = 15$$

   $$x = 3$$

As you can see, we arrive at the same value for x (3) using the substitution method. This reinforces the correctness of our previous solution using Cramer's Rule. The substitution method is particularly useful when one equation can be easily solved for one variable, making the substitution process straightforward.

Elimination Method: The Art of Cancellation

The elimination method (also known as the addition method) involves manipulating the equations so that the coefficients of one variable are opposites. Then, we add the equations together, which eliminates that variable, leaving us with a single equation in the other variable. Let's apply this to our system:

1. Multiply the first equation by 3 and the second equation by 2:

   $$-6x + 9y = -9$$

   $$6x - 8y = 10$$

2. Add the two equations together:

   $$y = 1$$

3. Substitute the value of y (1) back into either of the original equations to solve for x:

   $$-2x + 3(1) = -3$$

   $$-2x = -6$$

   $$x = 3$$

Again, we find that x = 3. The elimination method shines when the coefficients of one variable are easily made opposites by multiplication. It's a powerful technique for simplifying systems of equations and isolating variables.

Method Selection: Choosing the Right Tool for the Job

So, we've explored three different methods for solving systems of linear equations: Cramer's Rule, substitution, and elimination. Each method has its strengths and weaknesses, and the best choice often depends on the specific system you're dealing with. Cramer's Rule is elegant and provides a direct formula, but it can be computationally intensive for larger systems. Substitution works well when one variable is easily isolated. Elimination is effective when coefficients can be readily manipulated to cancel out variables. Becoming proficient in all three methods gives you the flexibility to tackle a wide range of problems and choose the most efficient approach.

Common Pitfalls and How to Avoid Them

Solving systems of equations can be tricky, and there are several common pitfalls that students often encounter. Let's highlight some of these and discuss how to avoid them:

• Sign Errors: One of the most frequent mistakes is making errors with signs, especially when dealing with negative numbers. Be extra careful when multiplying, distributing, or adding equations. Double-check your work and use parentheses to keep track of negative signs.
• Incorrectly Calculating Determinants: Cramer's Rule relies heavily on accurate determinant calculations. Remember the formula (ad - bc) and make sure you're multiplying the correct elements. A simple mistake here can throw off your entire solution. Practicing determinant calculations will significantly reduce this risk.
• Algebraic Errors: Mistakes in algebraic manipulation, such as combining like terms or simplifying expressions, can lead to incorrect results. Take your time, write out each step clearly, and double-check your algebra. Breaking down complex steps into smaller, manageable ones can also help prevent errors.
• Forgetting to Solve for Both Variables: In a system of two equations, you need to find the values of both x and y. Don't stop after solving for one variable; remember to substitute back into an equation to find the other variable. This ensures you have a complete solution.
• Not Checking Your Solution: The ultimate safeguard against errors is to check your solution. Substitute the values you found for x and y back into the original equations. If both equations are satisfied, your solution is correct. If not, you know there's an error somewhere, and you can go back and review your work. This simple step can save you a lot of points on exams!

Conclusion: Mastering the Art of Solving Equations

We've journeyed through the world of systems of linear equations, dissecting Cramer's Rule, exploring alternative methods like substitution and elimination, and highlighting common pitfalls to avoid. We discovered that the given expression accurately represents the setup for finding x using Cramer's Rule, but we needed to calculate the determinants to arrive at the actual value of x, which is 3. Solving systems of equations is a fundamental skill in mathematics, and mastering it opens doors to more advanced topics. By understanding the underlying principles, practicing different methods, and being mindful of potential errors, you can confidently tackle any system of equations that comes your way. So, keep practicing, keep exploring, and keep enjoying the beauty of mathematics!