Solving Systems Of Equations 8x + 3y = 24 And -4x - 7y = -12

In the realm of mathematics, solving systems of equations is a fundamental skill with applications spanning various fields, from engineering to economics. This article delves into a detailed, step-by-step approach to solving the specific system of equations:

$$\begin{array}{l} 8x + 3y = 24 \\ -4x - 7y = -12 \end{array}$$

We will explore the methods of substitution and elimination, providing a clear understanding of each technique and how to apply them effectively. By the end of this guide, you will not only be able to solve this particular system but also gain a solid foundation for tackling similar problems.

Understanding Systems of Equations

Before diving into the solution, it's crucial to grasp the concept of a system of equations. A system of equations is a set of two or more equations containing the same variables. The solution to a system of equations is the set of values for the variables that satisfy all equations simultaneously. In simpler terms, it's the point where the lines represented by the equations intersect on a graph.

In our case, we have two linear equations with two variables, x and y. Each equation represents a straight line, and our goal is to find the point (x, y) that lies on both lines. This point represents the solution to the system.

Method 1: The Elimination Method

The elimination method involves manipulating the equations in the system to eliminate one variable, allowing us to solve for the other. This is achieved by multiplying one or both equations by a constant so that the coefficients of one variable are opposites. When the equations are added, that variable is eliminated.

Step 1: Manipulating the Equations

Observe the given system:

$$\begin{array}{l} 8x + 3y = 24 \\ -4x - 7y = -12 \end{array}$$

Notice that the coefficients of x are 8 and -4. To eliminate x, we can multiply the second equation by 2 so that the coefficient of x becomes -8, which is the opposite of 8 in the first equation.

Multiply the second equation by 2:

$$2(-4x - 7y) = 2(-12)$$

This simplifies to:

$$-8x - 14y = -24$$

Now our system looks like this:

$$\begin{array}{l} 8x + 3y = 24 \\ -8x - 14y = -24 \end{array}$$

Step 2: Eliminating x

Add the two equations together. Notice how the x terms cancel out:

$$(8x + 3y) + (-8x - 14y) = 24 + (-24)$$

This simplifies to:

$$-11y = 0$$

Step 3: Solving for y

Divide both sides by -11 to isolate y:

$$y = \frac{0}{-11}$$

Therefore:

$$y = 0$$

Step 4: Substituting y to Find x

Now that we have the value of y, we can substitute it back into either of the original equations to solve for x. Let's use the first equation:

$$8x + 3y = 24$$

Substitute y = 0:

$$8x + 3(0) = 24$$

This simplifies to:

$$8x = 24$$

Divide both sides by 8:

$$x = \frac{24}{8}$$

Therefore:

$$x = 3$$

Step 5: The Solution

We have found that x = 3 and y = 0. Thus, the solution to the system of equations is the ordered pair (3, 0).

Method 2: The Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This reduces the system to a single equation with one variable, which can be easily solved.

Step 1: Solving for One Variable

Let's solve the first equation for x:

$$8x + 3y = 24$$

Subtract 3y from both sides:

$$8x = 24 - 3y$$

Divide both sides by 8:

$$x = \frac{24 - 3y}{8}$$

Step 2: Substituting into the Other Equation

Now, substitute this expression for x into the second equation:

$$-4x - 7y = -12$$

Substitute $x = \frac{24 - 3y}{8}$:

$$-4\left(\frac{24 - 3y}{8}\right) - 7y = -12$$

Step 3: Solving for y

Simplify the equation:

$$-\frac{24 - 3y}{2} - 7y = -12$$

Multiply both sides by 2 to eliminate the fraction:

$$-(24 - 3y) - 14y = -24$$

Distribute the negative sign:

$$-24 + 3y - 14y = -24$$

Combine like terms:

$$-11y = 0$$

Divide both sides by -11:

$$y = 0$$

Step 4: Substituting y to Find x

Now that we have the value of y, we substitute it back into the expression we found for x:

$$x = \frac{24 - 3y}{8}$$

Substitute y = 0:

$$x = \frac{24 - 3(0)}{8}$$

This simplifies to:

$$x = \frac{24}{8}$$

Therefore:

$$x = 3$$

Step 5: The Solution

Again, we find that x = 3 and y = 0. The solution to the system of equations is the ordered pair (3, 0).

Verifying the Solution

To ensure our solution is correct, we can substitute the values of x and y back into the original equations and verify that they hold true.

For the first equation:

$$8x + 3y = 24$$

Substitute x = 3 and y = 0:

$$8(3) + 3(0) = 24$$

$$24 + 0 = 24$$

$$24 = 24$$

This equation holds true.

For the second equation:

$$-4x - 7y = -12$$

Substitute x = 3 and y = 0:

$$-4(3) - 7(0) = -12$$

$$-12 - 0 = -12$$

$$-12 = -12$$

This equation also holds true. Therefore, our solution (3, 0) is correct.

Conclusion

We have successfully solved the system of equations:

$$\begin{array}{l} 8x + 3y = 24 \\ -4x - 7y = -12 \end{array}$$

using both the elimination and substitution methods. Both methods led us to the same solution, (x, y) = (3, 0). This exercise demonstrates the power and versatility of these algebraic techniques in finding solutions to systems of equations. Mastering these methods is crucial for further studies in mathematics and its applications in various fields.

Remember, practice is key to proficiency. Work through similar problems to solidify your understanding and build confidence in your problem-solving abilities. Systems of equations are a cornerstone of mathematical reasoning, and a strong grasp of these concepts will serve you well in your academic and professional pursuits.

Further Practice

To reinforce your understanding, try solving the following systems of equations using both the elimination and substitution methods:

1. $$\begin{array}{l} 2x + y = 7 \\ x - y = 2 \end{array}$$

2. $$\begin{array}{l} 3x - 2y = 8 \\ x + 4y = -2 \end{array}$$

3. $$\begin{array}{l} 5x + 3y = 16 \\ 2x - 5y = -11 \end{array}$$

By working through these examples, you'll not only become more comfortable with the mechanics of solving systems of equations but also develop a deeper intuition for choosing the most efficient method for a given problem. Whether you prefer the systematic approach of elimination or the directness of substitution, having both tools at your disposal will make you a more confident and capable problem solver.

Keep practicing, and you'll find that solving systems of equations becomes a natural and rewarding part of your mathematical journey.