Problem 14 - Algebra 1: Concepts and Skills

Question

Write in point-slope form the equation of the line that passes through the given points. $$ (-7,2) \text { and }(0,1) $$

Step-by-Step Solution

Answer

The equation of the line passing through the points (-7,2) and (0,1) in point-slope form is \(y = \frac{-1}{7}x + 3\).

Step 1: Calculate the Slope

Use the formula \(m= \frac{y_2 - y_1}{x_2 - x_1}\) to calculate the slope. Substitute the coordinates of the given points into the formula: \(m= \frac{1 - 2}{0 - (-7)} = \frac{-1}{7}\)

Step 2: Substitute into point-slope form

Next, substitute the slope and the coordinates of one of the points into the point-slope form of the line equation. It doesn't matter which point you choose; here, let's use (-7,2). The equation becomes: \(y - 2 = \frac{-1}{7}(x - (-7))\)

Step 3: Simplify the equation

Finally, simplify the equation to make it easier to read: \(y - 2 = \frac{-1}{7}x + 1\). If you add 2 to both sides, the equation simplifies to: \(y = \frac{-1}{7}x + 3\)

Key Concepts

Calculating Slope

Understanding how to calculate the slope is crucial when dealing with linear equations in algebra. The slope measures the steepness or inclination of a line and is represented by the symbol 'm'. The basic formula to calculate the slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) on a graph is \( m= \frac{y_2 - y_1}{x_2 - x_1} \).

When applying this formula, it's important to subtract the y-coordinate of the first point from the y-coordinate of the second point and do the same for the x-coordinates. The outcome will either be a positive or negative fraction, whole number, or zero. A positive slope indicates an upward trend from left to right, while a negative slope signifies a downward trend. A slope of zero means the line is horizontal; it does not rise or fall.

In our exercise, the slope is calculated by substituting the coordinates \( (-7, 2) \) and \( (0, 1) \) into our formula, yielding a slope of \( \frac{-1}{7} \), which tells us the line falls slightly as it moves from left to right.

Linear Equations

Linear equations form the foundation for understanding lines in algebra. A linear equation is an algebraic expression that represents a straight line when plotted on a graph. The most common form of a linear equation is the slope-intercept form, given as \( y = mx + b \) where 'm' is the slope and 'b' is the y-intercept—the point where the line crosses the y-axis.

Another form of the linear equation is point-slope form, which is especially useful when you have a point and the slope. It is written as \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is the point on the line and 'm' is the slope.

The exercise involves writing the equation of a line in point-slope form, which illustrates a direct application of understanding linear equations. By integrating the calculated slope and the coordinates of one of the provided points, we can construct the specific equation that describes the line through these points.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation (+, −, ×, ÷). In the context of linear equations, the variables and numbers express the relationship between the x and y coordinates on a graph.

To compose a linear equation from an algebraic expression, we need to incorporate known values, such as the slope and specific points, into the expression's general form. For example, after calculating the slope ('m') and selecting a point \( (x_1, y_1) \), we insert these into the point-slope form, which is an algebraic expression, to articulate the exact equation of our line.

Algebraic expressions can be simplified to ease the interpretation and further manipulation. For instance, in our solution, simplifying the point-slope equation \( y - 2 = \frac{-1}{7}x + 1 \) to \( y = \frac{-1}{7}x + 3 \) makes it more straightforward. Such simplification often involves combining like terms and making the equation more accessible for graphing or solving for specific values.