Problem 35 - Algebra 1

Question

Write an equation in point-slope form of the line that passes through the given point and has the given slope. $$ (-3,4), m=6 $$

Step-by-Step Solution

Answer

The equation of the line in point-slope form that passes through the point (-3,4) and has a slope of 6 is \(y - 4 = 6(x + 3)\)

Step 1: Identify the given point and slope

From the problem, we identify the given point as (-3,4) and the slope m as 6.

Step 2: Write down the point-slope form

The point-slope equation of a line is given as \(y - y_1 = m(x - x_1)\).

Step 3: Substitute the given values

Plug the values into the formula, replacing \(x_1\) with -3, \(y_1\) with 4, and m with 6, resulting in the equation \(y - 4 = 6(x - (-3))\).

Step 4: Simplify the resulting equation

Simplify the equation to its correct form, so the final equation is \(y - 4 = 6(x + 3)\).

Key Concepts

Equation of a Line

The equation of a line serves as a mathematical representation describing the relationship between all the points on a straight path. This can be expressed in various forms, notably including the point-slope form and slope-intercept form. Understanding these forms helps in graphing lines and solving geometrical problems easily.
To write an equation of a line, it usually involves some key components: a slope, which is the steepness of the line, and a point that the line passes through. Given a point \((-3, 4)\) and a slope of \(m = 6\), one can use these components to begin forming a line's equation.
Two common formats of line equations include:

• Point-Slope Form: Useful when a point and slope are known, written as \(y - y_1 = m(x - x_1)\).
• Slope-Intercept Form: Ideal for graphing, revealing both slope and y-intercept directly, written as \(y = mx + b\).

These forms serve unique purposes based on the information given and what you need to find.

Slope-Intercept Form

Slope-Intercept Form is one of the most popular ways to express linear equations, especially when graphing is involved. It is given by the formula \(y = mx + b\), where:

• \(m\) represents the slope of the line, showing how steep or flat the line is.
• \(b\) is the y-intercept, which is the point where the line crosses the y-axis.

This form is straightforward because it instantly indicates the slope and the y-intercept. The slope-intercept form is particularly useful when you need to quickly graph a line, as these two parameters can be plotted directly on a coordinate plane.
When converting from other forms, such as point-slope or standard form, to slope-intercept form, algebraic manipulations will show the value of \(b\), which can offer insights into the geometric positioning of the line.

Linear Equations

Linear equations are equations of the first order, which means each term is either a constant or the product of a constant and a single variable. The graph of a linear equation in two variables \(x\) and \(y\) is a straight line.
They are called 'linear' because they form a line when graphed. Examples of linear equation forms include:

• Standard Form: \Ax + By = C\, which is practical for algebraic manipulations.
• Point-Slope Form: \(y - y_1 = m(x - x_1)\), particularly useful when a point and slope are given.
• Slope-Intercept Form: \(y = mx + b\), ideal for graphing.

Recognizing these can help when translating between forms and solving for unknowns or plotting on a coordinate plane. The beauty of linear equations is their predictability and uniformity, making them easier to work with compared to higher-degree equations.