4. Graphing Linear Equations

The most common form is:

𝑦 = 𝑚𝑥 +𝑐

Where: • 𝑚 is the slope (rate of change) • 𝑐 is the y-intercept (where the line crosses the y-axis)

The slope-intercept form of a linear equation is a way of expressing the relationship between two variables, x and y, where y is dependent on x. The equation is structured as y = mx + c, where 'm' represents the slope of the line, indicating how steep the line is, and 'c' denotes the y-intercept, which is the point where the line crosses the y-axis. If you understand this format, you can quickly identify how changes in the value of x affect y. Got it 👍 Let me explain dependent and independent solutions for linear equations in one variable (Class 10 IB) and also extend to the special cases so it’s crystal clear:

🔹 Linear equation in one variable

A linear equation in one variable looks like:

where $a$ and $b$ are constants, and $x$ is the variable.

🔹 Independent solution (unique solution)

• When $a \neq 0$, the equation becomes:

  $$ax + b = 0 \quad \Rightarrow \quad x = -\frac{b}{a}$$

• This gives exactly one solution.

• Example: $2x + 6 = 0 \;\Rightarrow\; x = -3$.

🔹 Dependent solutions (infinitely many solutions)

• If both coefficients are zero, i.e. equation reduces to:

  $$0x + 0 = 0 \;\;\Rightarrow\;\; 0 = 0$$

• This is always true, no matter what $x$ is.

• Hence, infinitely many solutions (the equation is always satisfied).

• Example: $0x + 0 = 0$.

🔹 No solution (special case)

• If $a = 0$ but $b \neq 0$, the equation reduces to:

  $$0x + b = 0 \;\;\Rightarrow\;\; b = 0$$

• But since $b$ is not zero, this is impossible.

• Hence, the equation has no solution.

• Example: $0x + 5 = 0$ → $5 = 0$ (false).

Think of climbing a hill. The slope (m) would tell you how steep the hill is: a higher slope means a steeper hill, while a lower slope means a gentler incline. The y-intercept (c) tells you where you start climbing from on the y-axis, like starting at sea level before heading uphill.

1. Identify slope 𝑚 and y-intercept 𝑐.
2. Plot the point (0,𝑐) on the y-axis.
3. Use the slope 𝑚 = 𝑟𝑖𝑠𝑒/𝑟𝑢𝑛 to find the next point.
4. Draw a straight line through the points.

To graph the linear equation in slope-intercept form, follow these steps: First, determine the slope (m) and the y-intercept (c). If c is, for example, 3, you would plot the point (0, 3) on the y-axis. Next, use the slope to find another point; for a slope of 2 (which can be expressed as 2/1), you would rise 2 units up and run 1 unit to the right from the point you just plotted. Mark this new point, and repeat as necessary. Finally, connect these points with a straight line. This line will represent all the solutions to the equation.

Imagine you're planning a road trip. The y-intercept is your starting point at home (where you begin on the y-axis), and the slope represents how quickly you're traveling (rising in elevation at a rate of, say, 2 miles for every 1 mile you drive). As you plot your journey on a map, you'll connect points that illustrate how far you've traveled at each step.

--