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Chapter 2: Problem 46

Sketch the straight line defined by the linear equation by finding the \(x\) - and \(y\) -intercepts. $$ -2 x-8 y+24=0 $$

Short Answer

Expert verified
The x-intercept is found by setting \(y = 0\) in the equation, which yields \(x = 12\). The x-intercept is (12, 0). The y-intercept is found by setting \(x = 0\) in the equation, which yields \(y = 3\). The y-intercept is (0, 3). To sketch the line, plot these points and draw a straight line passing through them.

Step by step solution

01

Find the x-intercept

To find the x-intercept, set y = 0 and solve the equation for x: $$ -2x - 8(0) + 24 = 0 $$ Simplify the equation: $$ -2x + 24 = 0 $$ Now, solve for x: $$ x = \frac{24}{2} = 12 $$ The x-intercept is thus (12, 0).
02

Find the y-intercept

To find the y-intercept, set x = 0 and solve the equation for y: $$ -2(0) - 8y + 24 = 0 $$ Simplify the equation: $$ -8y + 24 = 0 $$ Now, solve for y: $$ y = \frac{24}{8} = 3 $$ The y-intercept is thus (0, 3).
03

Sketch the line

Now that we have the x-intercept (12, 0) and the y-intercept (0, 3), we can sketch the line. Plot the two points on a graph, and then draw a straight line that passes through them. That line represents the straight line defined by the given linear equation: $$ -2x - 8y + 24 = 0 $$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

X-Intercept Calculation
Understanding how to find the x-intercept of a linear equation is crucial when analyzing linear relationships. In simple terms, the x-intercept is where a straight line crosses the x-axis. To find the x-intercept, we set the y-coordinate to zero and solve for x.

For example, consider the equation (-2x - 8y + 24 = 0). Setting y to 0, the equation becomes (-2x + 24 = 0). Solving this for x gives us the x-intercept.

It's a straightforward process: subtract 24 from both sides to get (-2x = -24), and then divide both sides by -2, yielding (x = 12). Therefore, the x-intercept is at the point (12, 0). It's beneficial to note that the x-intercept represents a point where the line is at ground level, metaphorically speaking; it's where the output value is zero.
Y-Intercept Calculation
The y-intercept, analogous to the x-intercept, is where the line crosses the y-axis. To find it, we set x to zero and solve for y. In our example (-2x - 8y + 24 = 0), making x zero simplifies the equation to (-8y + 24 = 0).

Just like with the x-intercept, we solve for y by isolating the variable on one side. You add 8y to both sides and then divide by 8 to obtain (y = 3). Therefore, the y-intercept is at the point (0, 3). This tells us the height at which the line starts on the graph when x is zero, which can be particularly helpful for imagining how a line will look when sketching it.
Graphing Linear Equations
Graphing is a powerful tool for visualizing the relationship expressed by a linear equation. It involves plotting points and drawing a line that extends through them. With the x-intercept (12, 0) and the y-intercept (0, 3), these two points provide anchors for our line.

When sketching, you first plot these intercepts on a coordinate grid. The x-intercept is placed 12 units along the x-axis, and the y-intercept is placed 3 units up the y-axis. You then draw a line through these points which extends in both directions; this represents all the solutions of the equation (-2x - 8y + 24 = 0).

It's essential to ensure the line is straight and extends across the entire graph, as this demonstrates the continuous nature of linear relationships. Remember, every point on your drawn line is a solution to the equation. Learning to graph lines accurately aids in understanding functions and provides a basis for learning more complex concepts in algebra.

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