Problem 70 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find a system of linear equations that has the given solution. (There are many correct answers.) (-6,1)

Step-by-Step Solution

Answer

A possible system of linear equations that fits the solution \(-6,1\) could be: \[y = x + 7\] and \[y = 2x + 13\]. These are just an example, other systems of linear equations can also be generated using the same solution point, differing only by the chosen slopes and y-intercepts.

Step 1: Formation of First Equation

In order to form the first equation, it's ideal to pick arbitrary values for the slope and y-intercept. For example, use the point-slope form of the line equation \(y - y1 = m(x - x1)\), where \(m\) is the slope. For the sake of simplicity, choose \(m=1\). Substituting the point \(-6,1\) gives the first equation as: \(y - 1 = 1(x - (-6))\), which simplifies to: \(y = x + 7\).

Step 2: Formation of Second Equation

In a similar manner, the second equation is formed but this time, choose a different value for the slope, say \(m=2\). Using the same point \(-6,1\), the equation becomes: \(y - 1 = 2(x - (-6))\), which simplifies to: \(y = 2x + 13\).

Key Concepts

Understanding the Point-Slope Form

The point-slope form is incredibly useful for writing the equation of a line when you have the coordinates of one point on the line and the slope. It's expressed as y - y_1 = m(x - x_1), where (x_1, y_1) are the coordinates of the known point and m is the slope of the line.

This form offers a direct way to construct an equation based on key features of the line, making it a good starting point for problems where these aspects are provided. It can quickly transform into other forms, such as the slope-intercept form, by distributing the slope and moving terms around.

Let's illustrate with an exercise improvement: suppose we're given the point (-6, 1). Plugging this point into the point-slope formula with a slope of 1 yields y - 1 = 1(x + 6), simplifying this gives us the linear equation y = x + 7.

The Anatomy of a Linear Equation

A linear equation is a fundamental concept in algebra that shows the relationship between two variables, usually x and y, creating a straight line when graphed. The most general form of a linear equation is Ax + By = C, where A, B, and C are constants.

Linear equations are essential for solving systems, predicting trends, and understanding rates of change. Our example above, y = x + 7, is a linear equation displaying a direct relationship between x and y. Every value increase in x results in the same increase in y, clear evidence of a consistent slope.

The Concept of Slope

Slope is the measure of the steepness or incline of a line, reflecting how much y changes for a given change in x. It is denoted by m and can be calculated by the ratio of the rise over run—how much the line goes up for each unit it goes across. The formula for slope given two points (x_1, y_1) and (x_2, y_2) is m = (y_2 - y_1) / (x_2 - x_1).

Slope determines the angle and direction of a line on a graph. Positive slopes ascend from left to right, negative slopes descend, and a zero slope means the line is horizontal. By selecting different slopes, we can form different linear equations from the same point, as seen in our exercise solution, which shows equations for slopes of 1 and 2.

Y-intercept: Where the Line Meets the Y-Axis

The y-intercept is where the line crosses the y-axis on a graph. This is the value of y when x is zero. It's an integral part of the slope-intercept form of a linear equation, which is y = mx + b, where b is the y-intercept.

Understanding the y-intercept is valuable for graphing and interpreting lines, as it provides a specific point where the line will pass through. It is also vital when equations are used to model real-world situations because it often represents the starting or initial value. For instance, in our exercise solution, for the equation y = x + 7, the y-intercept is 7, which means the line touches the y-axis at the point (0, 7).