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The slope of a line is a measure of its steepness and direction. It is represented by the letter \(m\) in the equation of a line given in slope-intercept form: \(y = mx + c\). The slope tells us how much the \(y\)-value of a line increases or decreases as the \(x\)-value increases by 1 unit. If the slope is a positive number, the line slopes upwards from left to right. If it is negative, the line slopes downwards. A zero slope means the line is horizontal, and an undefined slope indicates a vertical line. In our exercise, the equation \(y = 14x\) shows us that the slope \(m\) is 14, meaning for every step to the right on the \(x\)-axis, the line rises 14 steps on the \(y\)-axis.
To summarize:

• Positive slope: Line rises as \(x\) increases.
• Negative slope: Line falls as \(x\) increases.
• Zero slope: Horizontal line.
• Undefined slope: Vertical line.

The y-intercept of a line is the point where it crosses the y-axis. In the slope-intercept form \(y = mx + c\), the \(c\) term represents the y-intercept. This point is crucial as it gives us the starting position of the line on the graph. In simple systems, knowing the y-intercept helps you quickly draw or visualise the line. In our equation \(y = 14x\), the y-intercept \(c\) is zero since no constant is explicitly added to the equation. This signifies that the line crosses the y-axis at the origin point \((0, 0)\).
Key takeaways:

• The y-intercept is where the line hits the y-axis.
• In \(y = mx + c\), \(c\) is the y-intercept, the value of \(y\) when \(x=0\).
• A zero y-intercept means the line passes through the origin.

Linear equations are mathematical expressions that describe a straight line. They are called "linear" because their graph is a straight line. The most commonly used form of linear equation is the slope-intercept form, \(y = mx + c\), which makes it easier to quickly understand the line's properties. The \(m\) indicates the slope, suggesting how the line moves, while the \(c\) tells you where the line crosses the y-axis. These equations are foundational in algebra and appear often in both academic studies and real-world applications.
In our exercise problem, \(y = 14x\) conforms to the slope-intercept form, thus making it straightforward to identify the slope and y-intercept. With \(m = 14\) and \(c = 0\), this specific line is steep and starts from the origin.
Essential points:

• Linear equations describe straight lines.
• They commonly appear in the form \(y = mx + c\).
• Understanding \(m\) and \(c\) helps in visualizing the line.