Investigation 3: Graphing Three Linear Equations in Three Variables

New Concept

A three-dimensional coordinate system is a space that is divided into eight regions by an $x$-axis, a $y$-axis, and a $z$-axis.

Why it matters

Algebra is the language used to describe relationships, and moving to three variables allows us to model the 3D world we live in. Mastering these systems is the first step toward solving complex problems in fields like engineering, physics, and computer graphics.

What’s next

Next, you’ll apply the substitution method to solve systems with three variables, finding the single point where three planes intersect in space.

A space divided into eight regions by an x-axis, a y-axis, and a z-axis. The z-axis is vertical, the y-axis is horizontal, and the x-axis appears to go into the page. Coordinates in this system are called ordered triples and are written in the form $(x, y, z)$. The graph of an equation like $Ax + By + Cz = D$ is a plane.

To plot the point $(3, 5, 4)$, you move 3 units along the x-axis, 5 units parallel to the y-axis, and finally 4 units up parallel to the z-axis. An ordered triple like $(-1, 2, -3)$ represents a single point in this 3D space, the potential intersection of three different planes. The origin of the system is the point where all three axes meet, with coordinates $(0, 0, 0)$.

Think of the corner of your room! The floor has two lines (x and y), and the wall's edge going up is the z-axis. This system helps us pinpoint locations in 3D space, not just on a flat map. Every point gets a unique $(x, y, z)$ address, which is the solution where three planes can intersect.

To solve a system of three equations, first isolate one variable in one equation. Then, substitute this expression into the other two equations to create a new system with only two variables and two equations. Solve this smaller system for those two variables. Finally, substitute those values back into an original equation to find the value of the third variable.

Given $x - 2y + z = 7$, solve for $x$ to get $x = 2y - z + 7$. Now substitute $(2y - z + 7)$ for $x$ in other equations. After substitution, you might get a 2-variable system like $y + 5z = 10$ and $2y - z = 4$. Solve this smaller system first. If you find $y=3$ and $z=2$ from the smaller system, plug them back into an original equation to find the value of $x$.

It's like a detective mission! Use one clue (equation) to get a description of a suspect (variable). Then, plug that description into your other two clues. This creates a simpler two-suspect mystery that’s much easier to crack! Once you solve that, you can find out who the last suspect is.

To graph a linear equation in three variables like $Ax + By + Cz = D$, you find its intercepts on each axis. To find the x-intercept, set $y=0$ and $z=0$ and solve for $x$. Do the same for the y-intercept (set $x=0, z=0$) and z-intercept (set $x=0, y=0$). Plotting these three points helps visualize the plane.

For the equation $2x + 4y + z = 8$, the x-intercept is $(4, 0, 0)$ because when $y=0$ and $z=0$, $2x=8$. The y-intercept for the same equation is $(0, 2, 0)$, because setting $x=0$ and $z=0$ gives $4y=8$. The z-intercept for $2x + 4y + z = 8$ is $(0, 0, 8)$, because if $x=0$ and $y=0$, then $z=8$.

How do you draw a giant, flat sheet (a plane) in 3D space? Just find where it pokes through each of the three axes! Find these three 'poke points' (intercepts), plot them, connect the dots to form a triangle, and you've created a neat sketch of your plane in space. It's that simple!

When beginning the substitution method, it is not necessary to always start with the first equation. To make the process simpler, you should carefully inspect all three equations and choose the one that is easiest to solve for a single variable. This is typically an equation where a variable, like $x$, $y$, or $z$, has a coefficient of 1 or -1.

In the system where one equation is $x + 4y - 5z = 7$, it is easiest to solve this equation for $x$ to get $x = 7 - 4y + 5z$. In a system with the equations $3x - 2y + 4z = 9$ and $2x + y - z = 1$, it is easiest to solve the second equation for $y$ or $z$. Given a choice between solving $3x+...=...$ for $x$ and $x+...=...$ for $x$, always choose the second one to avoid dividing everything by 3.

Work smarter, not harder! Before you start crunching numbers, scan your equations. If one of them has a lonely 'x' or 'y' without a number attached, pick that one! It’s your golden ticket to solving for a variable without making a mess of fractions, which saves you a ton of work later on.