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Understanding the equation of a line is key in algebra. A linear equation in two variables, such as \(2x - 3y = 12\), can be represented in different forms. The most adaptable form is the slope-intercept form, \(y = mx + b\). Here, \(m\) represents the slope of the line, while \(b\) is the y-intercept, where the line crosses the y-axis.Let's start by manipulating the original equation, \(2x - 3y = 12\). To write it in slope-intercept form, solve for \(y\). This might require rearranging the terms, isolating \(y\) on one side of the equation.

• Subtract \(2x\) from both sides: \(-3y = -2x + 12\)
• Divide every term by \(-3\) to solve for \(y\): \(y = \frac{2}{3}x - 4\)

The equation is now in the slope-intercept form, \(y = \frac{2}{3}x - 4\), where it becomes more straightforward to identify the slope and the y-intercept.

Graphing linear equations allows us to visually represent algebraic relationships between variables. Let's look at the equation \(y = \frac{2}{3}x - 4\). This is an example of a linear equation where the graph will be a straight line.To graph this equation:

• First, identify the y-intercept \((0, b)\). In our equation, \(b = -4\), so the y-intercept is at \( (0, -4)\).
• Plot this point on the graph as it shows where the line crosses the y-axis.
• Next, use the slope \(\frac{2}{3}\) to find another point. The slope tells us that for every 3 units moved horizontally to the right, the vertical change is 2 units up. Start from \( (0, -4)\), move 3 units to the right and 2 units up to reach the next point \( (3, -2)\).

Connect the y-intercept and the new point with a straight line. This visual representation clearly shows the linear relationship between \(x\) and \(y\). Extend the line in both directions with arrows to indicate it continues indefinitely.

The slope and y-intercept are essential components in defining the characteristics of a line in the slope-intercept form, \(y = mx + b\).

Slope

The slope, indicated by \(m\), determines the steepness and direction of the line. Positive slopes rise when moving left to right, while negative slopes fall. In our equation \(y = \frac{2}{3}x - 4\), the slope \(\frac{2}{3}\) means:

• For every increase of 3 units in \(x\), \(y\) increases by 2 units.
• The line rises gently compared to a steeper slope like \(2x\) (which would rise 2 units for every single unit increased in \(x\)).

Y-Intercept

The y-intercept, shown by \(b\), is the point \((0, b)\) where the line crosses the y-axis. It represents the value of \(y\) when \(x = 0\). From the equation \(y = \frac{2}{3}x - 4\), the y-intercept \(b\) is \(-4\), indicating the line crosses the y-axis at \((0, -4)\).These components fully describe the line's behavior in a two-dimensional plane, making them crucial for graphing and understanding linear equations.