Problem 46 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the slope and \(y\)-intercept of the equation of the line. Then sketch the line by hand. $$1.2 x+3.5 y=10.5$$

Step-by-Step Solution

Answer

The slope of the line is -1.2/3.5 and the y-intercept is 10.5/3.5. The line can be sketched by starting at the y-intercept and then using the slope to determine another point on the line and then drawing the line through these points.

Step 1: Transform the equation to the form y = mx + c

To do this, isolate y in the equation. By subtracting 1.2x from both sides we have: 1.2x - 1.2x + 3.5y = 10.5 - 1.2x, simplifying this leaves the equation of the line as: 3.5y = -1.2x + 10.5.

Step 2: Determine the slope and y-intercept

Divide the equation by 3.5 to get: y = (-1.2/3.5)x + (10.5/3.5). The coefficient of x in this equation is the slope (m), and the constant term is the y-intercept (c).

Step 3: Sketch the line

First plot the y-intercept on a graph. This is the point where the line crosses the y-axis. Using the slope (m), rise/run, determine a second point and draw a line through these two points.

Key Concepts

Linear Equations

Linear equations form the basis for understanding how variables interact in a straight-line relationship. An equation like the one provided, 1.2x + 3.5y = 10.5, describes a line in two-dimensional space and can be solved for either variable. In algebra, we often rearrange these equations to solve for y in terms of x, which makes the correlation between the variables more apparent. This particular equation represents a linear relationship because each term is either a constant or the product of a constant and a single variable raised to the first power.

In the world of algebra, the knowledge of manipulating linear equations is invaluable, as it allows students to explore how changes in one variable affect another and lays the groundwork for more advanced topics in mathematics and science. Seeing the direct proportionality or inverse relationship that variables may have is key in various real-world applications like physics, economics, and even social sciences.

Slope-Intercept Form

The slope-intercept form is a special way of writing the equation of a line so that it is particularly easy to graph. This form is written as y = mx + c, where m represents the slope of the line, and c represents the y-intercept, which is the point where the line crosses the y-axis.

To convert the given equation to slope-intercept form, algebraic manipulation is necessary to isolate y. As the solution shows, subtracting 1.2x from both sides of the original equation and then dividing by 3.5 isolates y and provides us with the slope (m = -1.2/3.5) and y-intercept (c = 10.5/3.5). This form is extremely useful for quickly sketching graphs and understanding how the line behaves without plotting multiple points.

Graphing Lines

Graphing lines is a visual way to represent the solution set of a linear equation. With the slope-intercept form, graphing becomes a two-step process. First, locate the y-intercept on the graph. For our equation, the y-intercept (c) is 10.5/3.5, so we would plot this point on the y-axis. Next, use the slope to find another point on the line. The slope tells us how to move from one point on the line to another. With a slope of -1.2/3.5, we'd move down 1.2 units and to the right 3.5 units from the y-intercept to find a second point. By connecting the y-intercept and this second point, we can draw the line.

This visual approach to understanding linear relationships is fundamental in fields that range from engineering to data analysis, as it provides immediate insight into trends and correlations.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions and equations to achieve a desired form or to isolate a particular variable. In the context of our given problem, this manipulation includes subtracting terms from both sides to get the y term by itself, and then dividing through by the coefficient of y to solve for y in terms of x. These techniques allow us to move fluently between different forms of equations, such as the standard form (Ax + By = C) and the slope-intercept form (y = mx + c), each of which has its uses depending on the context. Mastery of these skills is crucial for students as it enables them to solve a wide range of problems in both academic settings and real-life situations.