Question

Project: Solve the following set of equations. This will give us formulas for solving a set of three equations, which we will find useful in the next chapter. $$\begin{array}{l} a_{1} x+b_{1} y+c_{1} z=k_{1} \\ a_{2} x+b_{2} y+c_{2} z=k_{2} \\ a_{3} x+b_{3} y+c_{3} z=k_{3} \end{array} $$

Short answer

Solve the first equation for x, substitute that expression for x into the other two equations, simplify, solve the resulting two equations for y and z, and substitute those back in to find x.

Step-by-step solution

Express one variable in terms of the others

Choose one of the equations and solve it for one of the variables, for example, solve the first equation for x: \[ x = \frac{k_1 - b_1y - c_1z}{a_1} \]

Substitute the expression into the other equations

Substitute the expression for x into the second and third equations:\[ a_2\left(\frac{k_1 - b_1y - c_1z}{a_1}\right) + b_2y + c_2z = k_2 \]\[ a_3\left(\frac{k_1 - b_1y - c_1z}{a_1}\right) + b_3y + c_3z = k_3 \]

Simplify the equations

Multiply through by the denominators to eliminate fractions and simplify the equations from step 2:

Solve the simplified system for y and z

Now that we have two equations in two variables, solve for y and z using methods like substitution or elimination.

Substitute back to find x

After finding values for y and z, substitute these back into the expression for x obtained in step 1 to determine x.

Key concepts

Substitution Method

The substitution method is a technique used in algebra for solving systems of equations, where one of the equations is rearranged to express one variable in terms of the others, and then substituted into the other equations.

For example, given a set of simultaneous linear equations, you first solve one of the equations for one variable. If we have the equation \(a_1 x + b_1 y + c_1 z = k_1\), we might choose to express \(x\) in terms of \(y\) and \(z\) as \( x = \frac{k_1 - b_1y - c_1z}{a_1} \). You then substitute this expression into the remaining equations, which reduces the number of equations and variables you need to solve for.

After substitution, you will have simplified expressions that can be solved more easily, eventually finding the value for one variable and using it to find the others. Substitution is especially useful when the equations directly allow for easy isolation of one variable.

Elimination Method

The elimination method is another algebraic method for solving systems of equations where you aim to cancel out one variable by adding or subtracting the equations.

This method is particularly useful when the coefficients of one of the variables are opposites or can easily be made opposites. To apply elimination, you can multiply one or both of the equations by a number that allows the coefficients of one variable to become the negative of each other. Then, by adding or subtracting the equations, that variable will be eliminated.

For instance, if you have the equation system \( a_2 x + b_2 y + c_2 z = k_2 \) and \( a_3 x + b_3 y + c_3 z = k_3 \) after substitution, you might multiply the first by \(b_3\) and the second by \(b_2\) (considering that \(b_2\) and \(b_3\) are not zero), which will allow you to eliminate \(y\) after adding or subtracting the resulting equations. The remaining equation will contain only one variable, making it much simpler to solve.

Simultaneous Linear Equations

Simultaneous linear equations are a set of two or more algebraic equations with multiple variables where all equations are considered together at once. The term 'simultaneous' means that the solution must satisfy all given equations in the system at the same time.

These kinds of equations often emerge in real-world problems where more than one condition has to be satisfied concurrently. To solve these, one could use methods such as substitution or elimination. In the given exercise, we work with a set of three simultaneous equations, which form a three-dimensional geometric representation — where the solution is the point of intersection of three planes in space.

Each equation represents a plane, and the solution to the system is a point \( (x, y, z) \) that lies on all three planes at once. Finding this intersection point is the goal of solving simultaneous linear equations, and it requires systematic elimination of variables until all are determined.

Algebraic Methods

Algebraic methods encompass a variety of techniques used to solve equations, including simple linear equations, quadratic equations, systems of equations, and more complex algebraic structures.

The two primary methods for solving systems of equations are substitution and elimination, as mentioned earlier. These methods are foundational techniques in algebra and are widely applied across different fields of mathematics and sciences. In our context, these algebraic methods allow us to solve for \(x\), \(y\), and \(z\) in a system of three equations with three unknowns.

Algebraic methods involve manipulating equations and expressions using operations like addition, subtraction, multiplication, and division while respecting the properties of equality and equivalence. The aim is to arrange the equations into a more manageable form to systematically find the values of the unknown variables that satisfy all equations in the system. A strong understanding of these methods is essential for students pursuing studies in fields that require mathematical problem-solving skills.