Problem 50 - Algebra 1: Concepts and Skills

Question

Write in slope-intercept form the equation of the line that passes through the given point and has the given slope, or that passes through the given points. \((5,1), m=5\)

Step-by-Step Solution

Answer

The equation of the line that passes through the point (5,1) and has a slope of 5 is \(y = 5x - 24\).

Step 1: Insert the known values

To start the problem, we first insert the given reference point, which is \((5,1)\), and the given slope \(m=5\) into the slope-intercept form equation (\(y = mx+c\)). We get \(1 = 5 * 5 + c\).

Step 2: Solve for c

To solve for c in the equation \(1 = 25 + c\), subtract 25 from both sides. This means that \(c = 1 - 25 = -24\).

Step 3: Write the equation of the line

With the values of \(m\) and \(c\) known, we can now write the equation of the line that passes through the point (5,1) with a slope of 5, which is \(y = 5x - 24\).

Key Concepts

Linear Equations

Understanding linear equations is crucial to mastering algebra and advanced mathematics. A linear equation represents a straight line when plotted on a graph. The general form of a linear equation is 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. When given a point through which the line passes and the slope, we can plug these values into the slope-intercept form to find the equation.

For instance, if we know the line passes through the point (5,1) and has a slope of 5, we can start by substituting these values into the general form. This leads to an equation that must be solved for c, the y-intercept, to find the particular equation of this specific line. It is the y-intercept that allows us to uniquely identify the line among all lines that have the same slope.

Slope of a Line

The slope of a line is a number that describes both the direction and the steepness of the line. Calculating the slope is simple: it is the change in y (the vertical change) divided by the change in x (the horizontal change), often expressed as 'rise over run'. In the equation of a line in slope-intercept form, y = mx + c, the coefficient m represents the slope. A positive slope means the line rises as it moves from left to right, while a negative slope signifies that the line falls.

In our exercise, the given slope is 5, which implies that for every unit increase in x, y increases by 5 units. This slope can be visualized on a graph as a line that cuts across the coordinate plane at a steady incline. Using the slope helps us determine the general direction of the line, and when combined with a point on the line, allows us to derive the full linear equation.

Y-Intercept

The y-intercept is a fundamental aspect of the graph of a linear equation. It refers to the point on the graph where the line crosses the y-axis, which corresponds to the value of y when x is zero. In the slope-intercept form y = mx + c, the y-intercept is represented by c. This value is essential as it anchors the line at a specific location on the graph.

To find the y-intercept of the line in our exercise, we rearrange the equation to isolate c. By substituting the point (5,1) into the equation and solving for c, we establish that the y-intercept is -24. This means that if we were to draw the line on a coordinate plane, it would cross the y-axis at the point (0, -24). Knowing the y-intercept is particularly helpful when graphing the line or when wanting to quickly identify the point at which the line will pass through the y-axis.