Problem 24 - Algebra and Trigonometry

Question

Find an equation of the line that satisfies the given conditions. \(x\) -intercept \(-8 ; \quad y\) -intercept 6

Step-by-Step Solution

Answer

The equation of the line is \( 3x - 4y = 24 \).

Step 1: Understand Intercepts and Equation Form

To find the equation of a line given the intercepts, recall that a line with intercepts has the general form: \( \frac{x}{a} + \frac{y}{b} = 1 \), where \(a\) is the \(x\)-intercept and \(b\) is the \(y\)-intercept.

Step 2: Substitute Intercepts into Formula

Given \(x\)-intercept \(-8\) and \(y\)-intercept \(6\), substitute these values into the formula: \( \frac{x}{-8} + \frac{y}{6} = 1 \).

Step 3: Clear Fractions

Clear the fractions in the equation by multiplying every term by the least common multiple of the denominators, which is 24. The equation becomes: \( 3x - 4y = 24 \).

Step 4: Simplify the Equation

The equation \( 3x - 4y = 24 \) is already simplified and represents the line with the given intercepts.

Key Concepts

x-intercept

The x-intercept of a line is where the line crosses the x-axis. At this point, the value of y is zero because it's on the x-axis. Thus, you solve for the x-coordinate while the y-coordinate is zero. In the equation of a line with intercepts, the x-intercept is represented by 'a' in the formula:
\[ \frac{x}{a} + \frac{y}{b} = 1 \]

• Essentially, when you set y to 0 in the equation, you find when x equals the x-intercept.
• This gives a clear point on the graph, here it is (-8, 0).
• An important factor to remember is that the x-intercept can be negative, zero, or positive, depending on which direction the line crosses the x-axis.

To find the x-intercept from the equation, simply substitute y = 0, and solve for x.
In our example, substituting y = 0 in the equation \( 3x - 4y = 24 \), we end up confirming that at \( x = -8 \), the line crosses the x-axis.

y-intercept

The y-intercept is the point where the line crosses the y-axis, which occurs when the x-value is zero. Therefore, substitute x = 0 into your linear equation to uncover the y-intercept. The y-intercept is shown as 'b' in the intercept formula:
\[ \frac{x}{a} + \frac{y}{b} = 1 \]

• Setting x to 0 allows you to directly find the y-value where the line meets the y-axis.
• For this reason, the y-intercept is always of the form (0, b).
• Here, our y-intercept is (0, 6).
• The y-intercept tells you where the line will intersect with the vertical axis on the graph.

Understanding the y-intercept helps in graphing the line quickly and accurately. By plugging x = 0 into the original line equation \( 3x - 4y = 24 \), you'll verify that y = 6 when the line intersects the y-axis.

equation of a line

The equation of a line refers to the mathematical representation that describes every point on the line. For lines with specific intercepts, we use the formula:
\[ \frac{x}{a} + \frac{y}{b} = 1 \]where 'a' and 'b' are the x-intercept and y-intercept respectively.

• This equation allows a simple and efficient way to derive a line just from knowing where it crosses the axes.
• Given the intercepts, replace 'a' with the x-intercept and 'b' with the y-intercept to find the equation.
• For example, substituting \(-8\) for 'a' and \(6\) for 'b', we have \( \frac{x}{-8} + \frac{y}{6} = 1 \).
• Multiplying through by the least common multiple of the denominators often helps simplify the fractional form into a linear equation: here, it's \( 3x - 4y = 24 \).

This linear equation format is very helpful to graphically depict lines, predict intersections, and analyze linear relationships in a variety of mathematical contexts.