Solving Linear Equations: Find 2x + 3y + 4z Value

Hey guys! Today, we're diving into the fascinating world of linear equations. We've got a system of three equations with three unknowns (x, y, and z), and our mission is to not only find the values of these variables but also to calculate the value of the expression 2x + 3y + 4z. Sounds like a fun challenge, right? Let's break it down step by step.

Understanding the System of Equations

First, let's take a good look at the system of equations we're dealing with:

$$\begin{cases} 2x - y + z = 12 \\ 3x + 2y - z = -11 \\ x - 4y + 5z = 47 \end{cases}$$

What we have here is a set of three linear equations. Each equation represents a plane in 3D space, and the solution to the system is the point where all three planes intersect. Our goal is to find the coordinates (x, y, z) of this intersection point.

Methods to Solve Linear Equations

There are several methods we can use to solve a system of linear equations, such as:

• Substitution: Solving one equation for one variable and substituting that expression into the other equations.
• Elimination: Adding or subtracting multiples of the equations to eliminate one variable at a time.
• Matrix Methods: Using techniques like Gaussian elimination or finding the inverse of a matrix.

For this particular problem, the elimination method seems like a straightforward approach. We can strategically add or subtract the equations to eliminate variables and simplify the system.

Step-by-Step Solution

Let's get our hands dirty and solve this system using the elimination method.

Step 1: Eliminate z from the First Two Equations

Notice that the first equation has a +z term, and the second equation has a -z term. This is perfect for elimination! If we add these two equations together, the z terms will cancel out:

$$(2x - y + z) + (3x + 2y - z) = 12 + (-11)$$

Simplifying this, we get:

$$5x + y = 1$$

Let's call this new equation Equation (4).

Step 2: Eliminate z Again, This Time Using Equations (1) and (3)

To eliminate z again, we need to combine Equations (1) and (3) in a way that cancels out the z terms. We can multiply Equation (1) by -5 to get a -5z term, and then add it to Equation (3):

$$-5 * (2x - y + z) = -5 * 12 => -10x + 5y - 5z = -60$$

Now, add this modified equation to Equation (3):

$$(-10x + 5y - 5z) + (x - 4y + 5z) = -60 + 47$$

Simplifying, we get:

$$-9x + y = -13$$

Let's call this Equation (5).

Step 3: Solve the System of Two Equations (4) and (5)

Now we have a simpler system of two equations with two unknowns:

$$\begin{cases} 5x + y = 1 \\ -9x + y = -13 \end{cases}$$

We can eliminate y by subtracting Equation (5) from Equation (4):

$$(5x + y) - (-9x + y) = 1 - (-13)$$

Simplifying, we get:

$$14x = 14$$

Dividing both sides by 14, we find:

$$x = 1$$

Step 4: Substitute x Back into Equation (4) to Find y

Now that we know x = 1, we can substitute it back into Equation (4) to solve for y:

$$5(1) + y = 1$$

$$5 + y = 1$$

$$y = -4$$

Step 5: Substitute x and y Back into Equation (1) to Find z

We've found x = 1 and y = -4. Let's substitute these values back into Equation (1) to find z:

$$2(1) - (-4) + z = 12$$

$$2 + 4 + z = 12$$

$$6 + z = 12$$

$$z = 6$$

Step 6: Calculate 2x + 3y + 4z

Finally, we have the values: x = 1, y = -4, and z = 6. Now we can calculate the value of the expression 2x + 3y + 4z:

$$2(1) + 3(-4) + 4(6) = 2 - 12 + 24 = 14$$

The Answer

So, guys, after all that work, we've found that the value of 2x + 3y + 4z is 14. Awesome!

Key Takeaways

• Solving systems of linear equations is a fundamental skill in mathematics and has applications in various fields.
• The elimination method is a powerful technique for solving such systems.
• It's crucial to be organized and methodical when working through the steps.
• Always double-check your work to avoid errors!

Why This Matters

You might be wondering, why bother learning this stuff? Well, systems of linear equations pop up everywhere in the real world! They're used in:

• Engineering: Designing structures, analyzing circuits, and controlling systems.
• Economics: Modeling supply and demand, predicting market trends.
• Computer Graphics: Creating 3D models and animations.
• Data Science: Building machine learning models.

So, mastering these skills can open up a lot of doors!

Tips for Success

• Practice, practice, practice! The more you work through problems, the more comfortable you'll become with the techniques.
• Stay organized. Keep your work neat and clearly labeled to avoid mistakes.
• Check your solutions. Substitute your values back into the original equations to make sure they work.
• Don't be afraid to ask for help. If you're stuck, reach out to a teacher, tutor, or classmate.

Let's Keep Learning!

I hope this explanation was helpful, guys! Remember, mathematics is like a muscle – the more you exercise it, the stronger it gets. So, keep practicing, keep exploring, and keep learning!

If you have any questions or want to dive deeper into this topic, feel free to ask. Until next time, happy problem-solving!