🌐 Understanding Linearity

Introduction:

• Discover how to determine whether a system of equations is linear.
• Explore the fascinating connection between solutions algebraically and geometrically.

🌐 Understanding Linearity

In our journey through this course, a significant part of our focus will be on working with linear equations. These equations play a crucial role in various mathematical applications. So, what exactly is a linear equation?

📏 In Two Dimensions — A Line’s Story:

• When we’re dealing with two dimensions, a linear equation is like the instruction manual for drawing a line. Imagine a Cartesian plane — you’ve probably seen one with an x-axis and a y-axis.
• A linear equation in two dimensions has the form ax+by=c, where a, b, and c are constants (or just fancy math terms for fixed numbers). These constants guide the line’s behaviour on the graph.
• Figure 1.1 shows how all possible lines in a plane can be formed with appropriate values of a, b, and c.

Figure 1.1

play with the graph in desmos!!

general line in 2D

🎨 Graphically Speaking:

• Graphically, this equation paints the path of a line on the plane. The x and y in the equation represent the coordinates of points on the line.
• For example, if your equation is 2x+3y=6, it’s like saying, “Pick any x and y that satisfy this equation, and you’ll be on the line.” see Figure 1.2
• Figure 1.2 — shows three points that satisfy the equation 2x+3y=6. In green the point that sits on the y-axis, in blue we have the point that sits on the x-axis, in red we have a randomly chosen point that sits on the line.
• Note each combination of x and y values that satisfies the equation corresponds to a point on the line. For example, if we take the coordinates of the red point i.e. x = 1.87 and y = 0.73, and put them in the LHS(Left Hand Side) of the equation 2x+3y = 6 i.e.
  2(1.87) + 3(0.73)
  and solve it. like,
  2(1.87) + 3(0.73) = 5.93
  we get 5.93
  which is very close to the RHS of the equation i.e. 6. Leading us to LHS = RHS.
• This tells us that any point on the line will lead us to LHS = RHS.
• Note: we lose some precision in cutting off our x and y after 2 digits of decimal point and hence we get 5.93 instead of 6.

Figure 1.2

play around with the generalized graph in Desmos!!

Desmos | Graphing Calculator

🤔 Constants Unveiled:

• Let’s demystify a, b, and c. They are just numbers that give specific characteristics to the line.
• a and b determine the line’s slope (how steep it is), and c sets the line’s position in relation to the origin (where the line crosses the x- or y-axis).

🛤️ In a Nutshell:

• So, when we dive into solving and working with linear equations, remember that we’re essentially exploring the rules and behaviors of lines in the mathematical realm.
• Understanding these equations is like having a treasure map to explore the vast landscape of two-dimensional space.

➕ Beyond the Line:

• Now, as we broaden our horizons beyond two dimensions, linear equations evolve. In three dimensions, the equation morphs into the form ax+by+cz = d, capturing the essence of a plane rather than a line.

Fig 1.3 shows a plane constructed using eqn ax+by+cz = d

Fig 1.4 shows how any plane can be constructed using appropriate values of a, b, c and, d

Generalizing further, a linear equation with n variables adopts the structure a₁​x₁​+a₁x2​+…+an​xn ​= b. Here a1​,a2​,…,an​,b are constants, and x1​,x2​,…,xn​ are the variables, sometimes referred to as unknowns.

Play with a generalised plane in desmos!!

Plane

🔍 Exploring the Equation Landscape:

• So, as we navigate through linear equations with multiple variables, remember that we’re not just tracing lines on a plane. We’re now sculpting planes in three-dimensional space or even higher-dimensional spaces.
• It’s like wielding a mathematical paintbrush to capture the nuances of relationships between constants and variables.