Solving Systems Of Equations With Matrices: A Step-by-Step Guide
Hey guys! Ever felt lost in a maze of equations? Don't worry, we've all been there! But today, we're going to learn a super cool way to solve systems of equations using matrices. It might sound intimidating, but trust me, it's like unlocking a secret level in math! We'll break it down step-by-step, so you'll be a matrix master in no time. Let's dive into solving the following system of equations:
-x - 7y - z = -19
4x + 4y + 4z = 4
2x + y + 6z = 7
We aim to find the values of x, y, and z that satisfy all three equations simultaneously. Buckle up, because we're about to enter the world of matrices!
What are Matrices, Anyway?
Before we jump into solving, let's quickly chat about what matrices actually are. Think of a matrix as a neat little organized table of numbers. It's like a spreadsheet, but way more powerful! Matrices are used in all sorts of cool stuff, from computer graphics to engineering, and yes, even solving systems of equations!
Imagine a matrix as a grid, with rows going horizontally and columns going vertically. Each number in the grid is called an element. We can use these elements to represent our equations in a compact form. This is where the magic begins!
Setting Up the Matrix
Our first step is to transform our system of equations into a matrix. We'll create what's called an augmented matrix. This matrix will contain the coefficients of our variables (x, y, and z) and the constants on the right side of the equations.
Let's break it down:
- Coefficients of x, y, and z: These numbers are the multipliers in front of our variables. For example, in the first equation, the coefficient of x is -1, the coefficient of y is -7, and the coefficient of z is -1.
- Constants: These are the numbers on the right side of the equals sign. In our first equation, the constant is -19.
Now, let's arrange these into a matrix:
[ -1 -7 -1 | -19 ]
[ 4 4 4 | 4 ]
[ 2 1 6 | 7 ]
See how we've neatly organized the numbers? The vertical line separates the coefficients from the constants. This matrix is our starting point for solving the system.
Row Operations: Our Secret Weapon
Now comes the fun part! We're going to use row operations to manipulate our matrix. Think of row operations as allowed moves in a game. Our goal is to transform the matrix into a special form called row-echelon form (or even better, reduced row-echelon form). This form will make it super easy to read off the solution for x, y, and z.
There are three main types of row operations we can use:
- Swapping two rows: We can interchange any two rows in the matrix. It's like shuffling the order of our equations.
- Multiplying a row by a non-zero constant: We can multiply all the elements in a row by the same number (as long as it's not zero). This is like multiplying both sides of an equation by a constant.
- Adding a multiple of one row to another row: This is the most powerful operation! We can multiply one row by a constant and add the result to another row. This is like adding a multiple of one equation to another.
Our strategy is to use these row operations to create zeros in specific places in the matrix. We want to get a diagonal of 1s (from the top left to the bottom right) and zeros everywhere else in the coefficient part of the matrix. This might sound tricky, but we'll take it one step at a time.
Step-by-Step Solution using Row Operations
Let's get our hands dirty and solve the system! Remember, our augmented matrix is:
[ -1 -7 -1 | -19 ]
[ 4 4 4 | 4 ]
[ 2 1 6 | 7 ]
Step 1: Get a 1 in the top-left corner.
Currently, we have a -1 in the top-left corner. Let's multiply the first row by -1 to get a 1:
(-1) * [ -1 -7 -1 | -19 ] -> [ 1 7 1 | 19 ]
Our new matrix is:
[ 1 7 1 | 19 ]
[ 4 4 4 | 4 ]
[ 2 1 6 | 7 ]
Step 2: Get zeros below the 1 in the first column.
We want to eliminate the 4 in the second row and the 2 in the third row. To do this, we'll use the third row operation (adding a multiple of one row to another).
- To get a 0 in the second row, first column, we'll multiply the first row by -4 and add it to the second row:
(-4) * [ 1 7 1 | 19 ] + [ 4 4 4 | 4 ] -> [ 0 -24 0 | -72 ]
- To get a 0 in the third row, first column, we'll multiply the first row by -2 and add it to the third row:
(-2) * [ 1 7 1 | 19 ] + [ 2 1 6 | 7 ] -> [ 0 -13 4 | -31 ]
Our matrix now looks like this:
[ 1 7 1 | 19 ]
[ 0 -24 0 | -72 ]
[ 0 -13 4 | -31 ]
Step 3: Get a 1 in the second row, second column.
We have -24 in the second row, second column. Let's divide the entire second row by -24:
(1/-24) * [ 0 -24 0 | -72 ] -> [ 0 1 0 | 3 ]
Our matrix is now:
[ 1 7 1 | 19 ]
[ 0 1 0 | 3 ]
[ 0 -13 4 | -31 ]
Step 4: Get a 0 above and below the 1 in the second column.
- To get a 0 in the first row, second column, multiply the second row by -7 and add it to the first row:
(-7) * [ 0 1 0 | 3 ] + [ 1 7 1 | 19 ] -> [ 1 0 1 | -2 ]
- To get a 0 in the third row, second column, multiply the second row by 13 and add it to the third row:
(13) * [ 0 1 0 | 3 ] + [ 0 -13 4 | -31 ] -> [ 0 0 4 | 8 ]
Our updated matrix is:
[ 1 0 1 | -2 ]
[ 0 1 0 | 3 ]
[ 0 0 4 | 8 ]
Step 5: Get a 1 in the third row, third column.
We have a 4 in the third row, third column. Let's divide the third row by 4:
(1/4) * [ 0 0 4 | 8 ] -> [ 0 0 1 | 2 ]
Now our matrix looks like this:
[ 1 0 1 | -2 ]
[ 0 1 0 | 3 ]
[ 0 0 1 | 2 ]
Step 6: Get a 0 above the 1 in the third column.
We need to eliminate the 1 in the first row, third column. Multiply the third row by -1 and add it to the first row:
(-1) * [ 0 0 1 | 2 ] + [ 1 0 1 | -2 ] -> [ 1 0 0 | -4 ]
Our final matrix is:
[ 1 0 0 | -4 ]
[ 0 1 0 | 3 ]
[ 0 0 1 | 2 ]
Reading the Solution
Woohoo! We've reached the reduced row-echelon form! This matrix is super easy to interpret. Let's translate it back into equations:
- The first row tells us: 1x + 0y + 0z = -4, which simplifies to x = -4
- The second row tells us: 0x + 1y + 0z = 3, which simplifies to y = 3
- The third row tells us: 0x + 0y + 1z = 2, which simplifies to z = 2
So, our solution is x = -4, y = 3, and z = 2. We can write this as an ordered triple: (-4, 3, 2).
Final Answer: The Ordered Triple
Therefore, the solution to the system of equations is (-4, 3, 2). We did it!
Why Matrices are Awesome
Solving systems of equations using matrices might seem like a lot of work at first, but it's a powerful technique that works for any number of equations and variables. Plus, it's a really organized way to keep track of your calculations.
Matrices are also used in tons of other areas of math and science, so understanding them is a great investment in your mathematical journey. So keep practicing, and you'll be solving systems of equations like a pro in no time!