Problem 42 - Algebra 1

Question

Write the point-slope form of the equation of the line that passes through the point and has the given slope. Then rewrite the equation in slope-intercept form. $$ (-3,-1), m=4 $$

Step-by-Step Solution

Answer

The equation of the line in point-slope form is \(y + 1 = 4(x + 3)\). In slope-intercept form, the equation of the line is \(y = 4x + 11\).

Step 1: Write the point-slope form

This form is given by \(y - y_1 = m(x - x_1)\). Substitute \(x_1 = -3\), \(y_1 = -1\), and \(m = 4\) into the equation: \[y - (-1) = 4(x - (-3))\] which simplifies to: \[y + 1 = 4(x + 3)\]

Step 2: Expand the equation

Multiply out the right-hand side to get: \[y + 1 = 4x + 12\]

Step 3: Rewrite in slope-intercept form

Isolate y on one side by subtracting 1 from both sides, the equation becomes: \[y = 4x + 12 - 1\], which further simplifies to: \[y = 4x + 11\]

Key Concepts

Slope-Intercept Form

The slope-intercept form is one of the most widely used representations of a line in algebra. It's the first stop when understanding linear equations and takes on the intuitive format of y = mx + b, where m represents the slope of the line and b indicates the y-intercept—the point where the line crosses the y-axis.

Let's take the example from the exercise where we have a point (-3,-1) and a slope of 4. By rearranging the point-slope equation, we achieve the slope-intercept form as y = 4x + 11. This simplicity makes it incredibly powerful for graphing lines and performing algebraic manipulations; it cleanly shows the rate of change along the line (the slope) and the specific location where the line meets the y-axis (the intercept).

Students can improve their grasp of slope-intercept form by practicing converting equations from other forms to this format, noting how different values of m and b affect the line's orientation on a graph.

Linear Equations

In the world of algebra, linear equations are the straight lines of the equation universe, characterized by their constant rate of change. These are mathematical statements indicating that two expressions yield the same value for any values of their variables.

In the provided exercise, the expression y = 4x + 11 is a linear equation in slope-intercept form. The term 'linear' implies that when this equation is graphed on a coordinate plane, all the solutions will form a straight line. A linear equation will always have one or two variables raised only to the power of one and does not include variables multiplying each other.

Understanding linear equations is pivotal for students, as they are foundational in algebra and beyond. Practice can include solving for variables, graphing the equations to see their visual representations, and experimenting with how changes to the slope and y-intercept impact the graphed line.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations without an equality sign. This makes them different from equations, which equate two expressions. For example, 4(x + 3) from Step 2 of the exercise is an algebraic expression, not yet an equation.

Getting comfortable with these expressions is crucial, as they are the building blocks of algebra. Students often manipulate these expressions to simplify or solve equations. For instance, taking the original point-slope form and distributing the slope across the (x + 3) term, we transformed an algebraic expression into an easier-to-interpret linear equation.

Improving this concept comprehension can be achieved by practicing the expansion, factoring, and simplifying of various algebraic expressions. Mastery here lays the groundwork for solving more complex problems and understanding the language of algebra.