Problem 57 - Algebra 1: Concepts and Skills

Question

Write in point-slope form the equation of the line that passes through the given point and has the given slope. $$ (-3,6), m=\frac{1}{2} $$

Step-by-Step Solution

Answer

The equation of the line in point-slope form is \(y - 6 = \frac{1}{2}(x + 3)\).

Step 1: Identify the Given

The problem gives a point \((-3,6)\) and a slope \(m=\frac{1}{2}\). This means \(x_1 = -3\), \(y_1 = 6\), and \(m = \frac{1}{2}\).

Step 2: Substitute into the Point-Slope Form

Substitute the slope and point values into the point-slope form of a line equation. Remember, the formula for point-slope form is \(y - y_1 = m(x - x_1)\). So the equation becomes: \(y - 6 = \frac{1}{2}(x - (-3))\).

Step 3: Simplify the Equation

Simplify the equation: \(y - 6 = \frac{1}{2}(x + 3)\). If there were instructions to give the answer in slope-intercept form, this step would consist of further manipulations of the equation to reach that form. However, since the problem doesn't ask for slope-intercept form, there is no need to further simplify.

Key Concepts

Writing Equations of Lines

The ability to write the equation of a line in algebra is a fundamental skill that connects two important concepts: a line's slope and the coordinates of a point it passes through. Let's look at how we construct this equation. First, identify the slope of the line, denoted as \(m\), which measures the line's steepness or incline. Next, determine the coordinates of a point on the line—let's call this \( (x_1, y_1) \). With these, we can employ the point-slope form to form the equation of the line: \[ y - y_1 = m(x - x_1) \]. In our exercise, for a line with slope \( \frac{1}{2} \) passing through the point \( (-3,6) \), we plug these values into the point-slope form to get the equation. This form is particularly useful when you have a point and the slope, and it's a quick way to start sketching the behavior of the line in question.

Slope-Intercept Form

Another popular form of the equation of a line is the slope-intercept form, written as \[ y = mx + b \], where \(m\) represents the slope and \(b\) the y-intercept of the line—the point where the line crosses the y-axis. This form gives a clear picture of the line's slope and intercept just by looking at the equation. For example, if we wanted to express our previous point-slope equation in slope-intercept form, we would solve for \(y\) and simplify to find the value of \(b\). The slope-intercept form is heavily used because it makes graphing straightforward and is easily derived from the point-slope form with a bit of algebraic manipulation.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. In the context of writing line equations, we use algebraic expressions to describe the relationship between the x and y variables. For the given exercise, \( \frac{1}{2}(x + 3) \) is an algebraic expression representing the line's behavior as x changes. Understanding how to manipulate these expressions is critical in algebra. It allows us to simplify or rearrange equations, which is precisely what we do when converting between forms of line equations.

Linear Equations

At the heart of these topics are linear equations. These are equations where the highest power of the variable is 1, resulting in a graph that is a straight line. Whether we write them in point-slope, slope-intercept, or standard form (Ax + By = C), their graphed results are identical; they only offer different insights or are more suitable for certain calculations. Our exercise features a linear equation derived from given slope and point parameters. Mastery of linear equations is central to many areas of mathematics, including geometry and calculus, and forms the foundation for understanding more complex functions.