Problem 12 - Algebra Form and Function

Question

Write an equation in point-slope form for the line. Through (2,3) with slope \(m=5\)

Step-by-Step Solution

Answer

Answer: The equation of the line in point-slope form is \(y - 3 = 5(x - 2)\).

Step 1: Identify the given information

The exercise provides us with the point (2,3) and the slope \(m=5\). We will use the point-slope form formula, \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is the given point and \(m\) is the slope.

Step 2: Plug in the given values into the point-slope form formula

We need to plug in the point (2,3) into \((x_1, y_1)\) and the slope \(5\) into \(m\). So we get \(y - 3 = 5(x - 2)\).

Step 3: Simplify and write the final equation

Now, simply write down the equation obtained in the previous step: \(y - 3 = 5(x - 2)\). This is the equation of the line in point-slope form that passes through the point (2,3) and has a slope of 5.

Key Concepts

Understanding Linear Equations

Linear equations are the backbone of algebra, describing a straight line on a graph. These equations are expressed in the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are real numbers, and \(x\) and \(y\) are variables. In simpler terms, if you input a value for \(x\), you get a corresponding \(y\) value that fits on the line. This relationship forms a straight line when plotted on a coordinate plane.
Understanding how to manipulate and rearrange these equations can help solve numerous algebra problems. For instance, converting a point-slope form equation into other forms, like slope-intercept or standard form, can make it easier to graph and understand.
When approaching linear equations, remember:

• They graph as straight lines.
• The slope \(m\), dictates the line's steepness and direction.
• One point is enough to define the line alongside the slope.

Decoding the Slope-Intercept Form

The slope-intercept form is another way to express linear equations, written as \(y = mx + b\). This form is particularly user-friendly because it directly shows the slope \(m\) and the y-intercept \(b\). The y-intercept is the point where the line crosses the y-axis, which happens when \(x = 0\).

This form allows for quick graphing and easy understanding of the line's properties. When you have an equation like \(y = 5x + 1\), it tells you:

• The slope \(m\) is 5, indicating the line rises five units for every unit it moves to the right.
• The y-intercept \(b\) is 1, meaning the line crosses the y-axis at \(y = 1\).

To convert from point-slope form to slope-intercept form, simply solve for \(y\) by expanding the equation and rearranging terms. It highlights how versatile linear equations can be, adapting to different scenarios with ease.

Solving Algebra Problems with Understanding

Algebra problems often involve manipulating equations to find unknown variables. When working with linear equations, this usually involves determining either the slope or specific points on the line.
A solid grasp of basic algebraic techniques, such as isolating variables and substituting given values, is crucial. These skills help in solving and understanding algebra problems more effectively.
A step-by-step approach is beneficial when tackling algebra problems:

• Identify what you know, such as given points or slopes.
• Decide which equation or formula is most appropriate for the problem.
• Substitute the known values into this equation.
• Rearrange and solve for the unknown variable.

Whether you're converting equations between different forms or solving for missing values, practicing these steps will enhance your algebraic skills. Approach each problem patiently, knowing that these processes become more intuitive with practice.