Problem 55 - Elementary Algebra

Question

Find the coordinates of two points on the given line, and then use those coordinates to find the slope of the line. $$2 x-y=-7$$

Step-by-Step Solution

Answer

Slope is 2, points can be (0, 7) and (1, 9).

Step 1: Convert the Equation to Slope-Intercept Form

Rewrite the equation in the form of \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Starting with the equation \(2x - y = -7\), add \(y\) to both sides and then add 7 to both sides to get \(y = 2x + 7\). This is now in the desired slope-intercept form.

Step 2: Choose Two Points on the Line

Select any two values of \(x\) and find the corresponding \(y\) values using the equation \(y = 2x + 7\). For example, if \(x = 0\), then \(y = 2(0) + 7 = 7\), so one point is \((0, 7)\). If \(x = 1\), then \(y = 2(1) + 7 = 9\), so another point is \((1, 9)\).

Step 3: Use the Coordinates to Find the Slope

Use the formula for the slope \(m = \frac{y_2 - y_1}{x_2 - x_1}\) with the points identified in Step 2, \((0,7)\) and \((1,9)\). Substituting these into the formula gives \(m = \frac{9 - 7}{1 - 0} = \frac{2}{1} = 2\).

Step 4: Conclusion

The slope of the line given by the equation \(2x - y = -7\) is 2. The process involved rewriting the equation, choosing arbitrary points, and calculating the slope using a standard formula.

Key Concepts

Coordinates

Coordinates are the numerical values that represent a point's position on a graph. Typically, these are written as pairs \(x, y\), where \(x\) is the value along the horizontal axis, and \(y\) represents the position along the vertical axis. These values help in pinpointing exact locations of points on a two-dimensional plane. For example, in the problem at hand, we've derived two points: \(\(0,7\)\) and \(\(1,9\)\). Each of these pairs tells us how far along and how high up a particular point is located on our graph.

• In the coordinate \(\(0,7\)\), \(x = 0\) indicates the point lies directly on the vertical y-axis.
• In contrast, \(x = 1\) in \(\(1,9\)\) signifies the point is shifted one unit to the right from the y-axis.

Understanding how to determine and use these coordinates to find information, like the slope of a line, creates a solid foundation in analyzing and interpreting linear equations on a graph.

Slope-Intercept Form

The slope-intercept form of a linear equation is a way of expressing the equation of a line so that the slope and y-intercept are immediately apparent. This form is typically written as \(y = mx + b\), where:

• \(m\) is the slope of the line, representing how steep the line is.
• \(b\) is the y-intercept, the point where the line crosses the y-axis.

Converting the original equation \(2x-y=-7\) into this format involves rearranging it into \(y = 2x + 7\). This transformation highlights the slope, 2, and the y-intercept, \(b = 7\).
The slope \(2\) tells us that for every unit increase in \((x)\), \(y\) increases by 2 units.
Understanding the slope-intercept form allows you to easily identify key features of the line graph without the need to further manipulate the equation.

Linear Equations

Linear equations graph as straight lines and are foundational to understanding relationships in algebra. These equations are typically expressed in the form \(Ax + By = C\) or can be rearranged into the slope-intercept form, \(y = mx + b\). Each format has its own unique benefits in problem-solving.

• Linear equations involve variables raised only to the first power, indicating their graph will always be a straight line.
• Their solutions are points on this line, representing values that satisfy the equation.

The equation provided in the exercise \(2x-y=-7\) is a prime example of a linear equation. Solving such equations involves finding points, determining slopes, and, ultimately, deducing the line's characteristics on a graph.
By familiarizing oneself with linear equations and their properties, one can efficiently solve various mathematical and real-world problems that involve linear relationships.