91Ó°ÊÓ

To solve a linear equation, the main goal is to isolate the variable. In the equation provided, \(-3x + 40y = 120\), we can solve for either variable by plugging in specific values.
When we substitute a value for one variable, it allows us to solve the equation for the other variable.
For example, to find \( y \) when \( x = 0 \), we substitute as follows:
\(-3(0) + 40y = 120\).
This simplifies to \(40y = 120\).
Divide both sides by 40:
\( y = 3 \).
Similarly, by following similar steps for other values, we can find the corresponding solutions for \( x \) and \( y \).
This isolation strategy is crucial in solving any linear equation.

The coordinate plane is a two-dimensional surface where we can plot points, lines, and curves to represent mathematical relationships.
It consists of two axes: the horizontal axis (x-axis) and the vertical axis (y-axis).
Each point on this plane is represented by an ordered pair \( (x, y) \).
For example, the point \( (0, 3) \) means that \( x = 0 \) and \( y = 3 \).
In the context of graphing an equation, plotting these points helps in visualizing the relationship between \( x \) and \( y \).
Once points are plotted, a line can be drawn through them to represent the equation interdependently.
When graphing, ensure that your axes are clearly labelled and that you plot each point accurately according to its coordinates.

A table of solutions lists the values of \( x \) and \( y \) that satisfy the equation.
By plugging in values as shown in the previous steps, we obtain a complete table:

• \( (0, 3) \)
• \( (-40, 0) \)
• \( (40, 6) \)

These points are the solutions of the given equation.
The process of filling in this table involves solving the equation for one variable as we insert values for the other variable.
This step-by-step tabulation is helpful because it lays out clear points that can be used for graphing, making the overall solution much easier to visualize and understand.

The slope-intercept form of a linear equation is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
This form makes it easy to graph the equation because it provides direct information about the slope and where the line crosses the \( y \)-axis.
For the equation \(-3x + 40y = 120\), it can be rewritten in slope-intercept form.

• First, isolate \( y \) on one side: \(40y = 3x + 120\).
• Divide by 40: \( y = \frac{3}{40}x + 3 \).

Here, the slope \( m = \frac{3}{40} \) and the y-intercept \( b = 3 \).
This form is particularly useful when you need to quickly sketch the graph, as you can easily identify the starting point and the steepness of the line.