Question

Solve the system of equations. $$\left\\{\begin{aligned} x+2 y-7 z &=-4 \\ 2 x+y+z &=13 \\ 3 x+9 y-36 z &=-33 \end{aligned}\right.$$

Short answer

The solution to the system of equations is: x = 5, y = -8 - x = -8 - 5 = -13, and z = 1.

Step-by-step solution

Create 2 new equations

In order to simplify the given system of equations, two new equations are formed by subtracting second and third equation from the first one respectively.\n Let's define:\n Equation (4): First equation - Second equation \n Equation (5): First equation - Third equation. This gives:\n (4) \(-x + y - 8z = -17\)\n (5) \(-2x - 7y + 29z = 29\)

Simplify Equation (5)

Multiply Equation (5) by 2 to simplify it further, yielding:\n \(-4x - 14y + 58z = 58\)

Find z

Subtract Equation (4) from Equation (5) (i.e. Equation (5) - Equation (4)) to find z. This gives \(-3x - 15y + 66z = 75\), which simplifies to z = 1.

Substitute value of z

Substitute the value of z (which we found as 1) back into Equation (4) and solve for y: \n \(-x + y - 8*1 = -17\), which simplifies to y = -8 - x.

Substitute y and z in the original equation

Substitute the values of y and z which were found, into the original first equation: \n \(x + 2*(-8 - x) - 7*1 = -4\), which simplifies to:\n \(x -2x +16 - 7 = -4\), which further simplifies to x=5.

Key concepts

Linear Algebra

Linear algebra is a significant field of mathematics that deals with vectors, vector spaces, linear mappings, and systems of linear equations. It's foundational for understanding many mathematical concepts and is widely applicable in natural sciences, engineering, computer science, economics, and more.

In the context of solving systems of equations, linear algebra emphasizes working with multiple linear equations at once. These equations represent relationships between variables, where each variable is scaled by a constant and summed to equal a constant. By combining and manipulating these equations meaningfully, we can find solutions that satisfy all given constraints.

For example, in the original exercise, we are given a system of three linear equations with three variables, which formulates a problem that can be approached using various linear algebra methods such as substitution or elimination. Understanding these algebraic structures and methods is crucial for anyone venturing into advanced fields that rely on data, optimization, and modeling.

Substitution Method

The substitution method is one of the primary techniques used to solve systems of linear equations. It involves isolating one variable in one equation and then substituting that expression into the other equations. This allows you to reduce the number of variables and equations step by step until you find the values for all the unknowns.

Let's consider the original exercise, where we identified the value of z using the elimination method. After finding z, we used substitution by inserting that value back into another equation to find y. We then substitute both known values of z and y into another equation to find x. The substitution method is especially useful when you can easily isolate a variable, making it a perfect complement to other techniques such as elimination.

Elimination Method

The elimination method is a strategic approach to solving simultaneous equations by eliminating one variable at a time to eventually solve for the remaining variables. This method often involves adding or subtracting whole equations or multiplying them by constants to cancel out a variable.

In the step-by-step solution to our problem, we started by creating two new equations from the original three to simplify the system. We manipulated these equations to eliminate x and y, leaving an equation with just z. Once z was found, we eliminated z from another equation to find y, and so on, which eventually led us to the solution for all variables. The elimination method is particularly efficient when the equations are set up in a way that allows for easy elimination of variables.

Simultaneous Equations

Simultaneous equations are a set of equations containing multiple variables that are all true at the same time. The primary objective when working with simultaneous equations is to find values for the unknowns that make all equations valid concurrently. Systems of simultaneous equations can be represented in various forms such as two linear equations in two variables, or in our case, three linear equations in three variables.

The given exercise required solving a system of simultaneous equations. To tackle such problems, methods like substitution or elimination are customary. These equations often model real-world scenarios where multiple conditions must be satisfied at once, which is why mastering the art of solving them is crucial for translating mathematical understanding into practical solutions.