Problem 46 - Algebra 1

Question

Write an equation of the line that passes through the points. (5,2),(4,3)

Step-by-Step Solution

Answer

The equation of the line that passes through the points (5,2) and (4,3) is \(y = -x + 7\).

Step 1: Calculate the slope

The slope of a line passing through points \((x1, y1)\) and \((x2, y2)\) can be calculated using the formula \[m = \frac{{y2 - y1}}{{x2 - x1}}\]. Substitute the given points into the formula: \(m = \frac{{3 - 2}}{{4 - 5}} = -1\).

Step 2: Write the equation in point-slope form

The point-slope form of the equation of a line is \(y - y1 = m(x - x1)\). Substitute one of the given points and the calculated slope into the formula. For example, using the point (5,2) leads to: \(y - 2 = -1(x - 5)\).

Step 3: Transform to slope-intercept form

Simplify the equation to the slope-intercept form \(y = mx + b\). Distribute the slope (-1) in the equation: \(y - 2 = -x + 5\). Then solve for y to get: \(y = -x + 7\).

Key Concepts

Understanding Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most common ways to express lines. It is written as \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept. This form is incredibly useful for quickly identifying the rate of change and the starting point of a line on a graph.

The slope, \(m\), indicates the steepness of the line and can be any real number, positive, negative, or zero. A positive slope means the line rises as it moves from left to right, while a negative slope means it falls. A zero slope corresponds to a horizontal line.

The y-intercept \(b\) is the point where the line crosses the y-axis. It shows the value of \(y\) when \(x = 0\). Recognizing this point as soon as possible can help in graphing the line and understanding its behavior quickly.

In our original problem, after calculating the slope and using point-slope form, we converted the equation to slope-intercept form to find \(y = -x + 7\). Here, the slope \(-1\) implies the line descends, and the y-intercept is 7, meaning the line crosses the y-axis at that point.

Exploring Point-Slope Form

The point-slope form is another essential way to write the equation of a line. It is written as \(y - y_1 = m(x - x_1)\), where \(m\) is the slope, and \((x_1, y_1)\) is a specific point on the line.

This form is particularly handy when you know a point on the line and the slope. You simply plug in these values into the formula and instantly have a descriptive equation of the line. This approach simplifies writing the equation without needing to first find the y-intercept. Choose either point given to plug into the formula.

In our step-by-step solution, we used the point \((5,2)\) along with the slope \(-1\), to find the equation \(y - 2 = -1(x - 5)\). This form highlights the relationship between the line's inclination and a specific location on the line.

Translating from point-slope form to slope-intercept is straightforward: distribute the slope and solve for \(y\) to see the equation's slope and y-intercept more clearly.

The Process of Calculating Slope

Calculating the slope \(m\) of a line is a fundamental step in understanding linear equations. The slope is the ratio of the vertical change to the horizontal change between two points on a line, often remembered as "rise over run."

To find the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\), use the formula:

• \(m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\)

This formula calculates the difference in the y-values (rise) divided by the difference in the x-values (run).

In the exercise, we had points \((5,2)\) and \((4,3)\). Plugging into the slope formula, \(m = \frac{3-2}{4-5} = -1\), we find the slope to be \(-1\). This negative slope means as we move from left to right, the line goes downward.

Understanding how to calculate the slope helps you in writing equations in both slope-intercept and point-slope forms and is key to graphing and analyzing lines.