In Exercises 11–38, use the given conditions to write an equation for each line in point-slope form and slope-intercept form. x-intercept = -1/2 and y-intercept = 4

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1. Identify the x-intercept and y-intercept given in the problem: x-intercept is \(-\frac{1}{2}\) and y-intercept is 4.
2. Use the intercepts to find the slope \(m\) of the line. The formula for slope using intercepts is \(m = \frac{y_2 - y_1}{x_2 - x_1}\), where \((x_1, y_1)\) and \((x_2, y_2)\) are the intercepts.
3. Substitute the intercepts into the slope formula: \(m = \frac{4 - 0}{0 - (-\frac{1}{2})}\).
4. Write the equation in point-slope form using one of the intercepts, say the y-intercept \((0, 4)\): \(y - y_1 = m(x - x_1)\).
5. Convert the point-slope form to slope-intercept form \(y = mx + b\) by solving for \(y\).

Key Concepts

Point-Slope Form

Point-slope form is a way to express the equation of a line using a specific point on the line and its slope. The formula is given by y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope. This form is particularly useful when you know a point on the line and the slope, allowing for easy graphing and analysis.

Slope-Intercept Form

Slope-intercept form is another way to express the equation of a line, defined as y = mx + b, where m is the slope and b is the y-intercept. This form is advantageous because it directly shows the slope and where the line crosses the y-axis, making it easy to graph and understand the line's behavior in relation to the axes.

Intercepts

Intercepts are the points where a line crosses the axes of a graph. The x-intercept is the point where the line crosses the x-axis (y=0), and the y-intercept is where it crosses the y-axis (x=0). Knowing the intercepts allows for the quick construction of the line's equation and provides insight into its position and slope.