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Linear equations are equations that make straight lines when graphed on a coordinate system. They typically look like this:

\( ax + by = c \)
In our example, the equation is \( x - 4 = 0 \).
This equation can be rewritten in a standard linear form:
\( x = 4 \)
Linear equations can have one or more variables, but their highest exponent will always be one. That’s why they create straight lines on a graph.
They are easy to work with because they follow predictable patterns, which is helpful for identifying intercepts.

Solving equations means finding the values of the variables that make the equation true.
There are simple steps to solve linear equations:

• Identify the type of equation
• Manipulate the equation to isolate the variable
• Apply arithmetic operations to solve for the variable

In this exercise, we started with the equation \( x - 4 = 0 \).
To find the x-intercept, we quickly solved it as follows:
\[ x - 4 = 0 \]
We add 4 to both sides to isolate x:
\[ x = 4 \]
This tells us that when y is 0, x is 4. So, the intercept is (4,0).
For the y-intercept, we would typically set x to 0 and solve for y. But our equation does not include a y term, meaning there is no y-intercept.

A coordinate system helps us visually represent equations and their solutions on a graph.
The most common type is the Cartesian coordinate system, which uses two perpendicular lines called axes:

• The x-axis, which runs horizontally
• The y-axis, which runs vertically

Each point on the graph is defined by an ordered pair (x, y).
For our equation \( x - 4 = 0 \), we found the point (4,0) is the x-intercept.
This means that the graph of the equation crosses or touches the x-axis at this point.
If there is an intercept on the y-axis, it would mean the graph touches or crosses at (0, y), but in our example, such an intercept doesn't exist.
Understanding the coordinate system can make it much easier to see relationships between numbers and spot patterns.