Problem 74 - Elementary Algebra

Question

For the following problems, determine the slope and \(y\) -intercept of the lines. Round to two decimal places. $$ 8.09 x+5.57 y=-1.42 $$

Step-by-Step Solution

Answer

Answer: The slope of the line is -1.45, and the y-intercept is -0.25.

Step 1: Rewrite the equation in slope-intercept form

To rewrite the equation in slope-intercept form, we need to isolate \(y\) on one side. The given equation is: $$8.09x+5.57y=-1.42$$ To isolate \(y\), first move the \(8.09x\) term to the other side by subtracting \(8.09x\) from both sides: $$5.57y=-8.09x-1.42$$ Now, divide both sides by \(5.57\) to get \(y\) by itself: $$y=\frac{-8.09x}{5.57}-\frac{1.42}{5.57}$$

Step 2: Simplify the equation's slope and y-intercept

Now, we want to simplify the equation we found in the last step to find the slope and y-intercept. The simplified equation is: $$y=\frac{-8.09}{5.57}x-\frac{1.42}{5.57}$$ Calculating the values, we get: $$y=-1.45x-0.25$$

Step 3: Identify the slope and y-intercept

Now that we have the equation in the slope-intercept form, we can easily identify the slope (\(m\)) and the y-intercept (\(b\)). From our equation, we can see that the slope, \(m\) is \(-1.45\), and the y-intercept, \(b\) is \(-0.25.\) So, the slope of the line is -1.45, and the y-intercept is -0.25.

Key Concepts

Slope Calculation

Understanding how to calculate the slope of a line can be empowering in solving linear equations. When given a linear equation, it might not immediately appear in its most helpful form for recognizing the slope. A typical way to display the equation of a line is the slope-intercept form, which is easily recognizable as \(y = mx + b\), where \(m\) represents the slope.

The slope signifies the steepness and direction of the line on a graph. Steepness is expressed as a fraction, a ratio between the vertical change and the horizontal change between two points on the line, also known as "rise over run." For example, in our problem, the step-by-step solution was derived from the equation \(8.09x + 5.57y = -1.42\). By rearranging this equation to solve for \(y\), we identified the slope to be \(-1.45\).

The calculation involved turning \(8.09x\) into the form \(-\frac{8.09}{5.57}x\), revealing the slope as \(-1.45\). The resulting negative slope indicates a downward diagonal direction for the line across the graph.

Y-Intercept

The y-intercept is the point where the line crosses the y-axis on a graph. This concept is crucial as it provides a starting reference point for graphing the equation.

In any linear equation expressed in slope-intercept form \(y = mx + b\), the "b" represents the y-intercept. In our example, after converting the given equation to \(y = -1.45x - 0.25\), the y-intercept is visible as \(-0.25\). This means that the line will intersect the y-axis at the point \((0, -0.25)\).

Knowing the y-intercept is particularly useful if you need to quickly plot the graph of the line starting from this point and using the slope to guide the line's direction and steepness.

Linear Equations

Linear equations are foundational in algebra and form the backbone of many real-world applications, such as predicting trends and creating models. A linear equation represents a relationship in which each value of \(x\) correlates directly with a corresponding value of \(y\), following a constant rate of change.

Typically, these equations are represented in the form \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants. The ultimate goal when dealing with such equations in problems like ours is to rearrange this standard form into the easy-to-read slope-intercept format \(y = mx + b\).

This transformation to slope-intercept form allows both the slope and y-intercept to be easily identified. Understanding linear equations and their properties enables us to graph lines accurately, solve systems of equations, and apply these skills to various fields, such as economics, physics, and engineering.