Problem 1 - Algebra and Trigonometry Real Mathematics, Real People

Question

Match each equation with its form. (a) \(A x+B y+C=0\) (i) vertical line (b) \(x=a\) (ii) slope-intercept form (c) \(y=b\) (iii) general form (d) \(y=m x+b\) (iv) point-slope form (e) \(y-y_{1}=m\left(x-x_{1}\right)\) (v) horizontal line

Step-by-Step Solution

Answer

(a)-(iii), (b)-(i), (c)-(v), (d)-(ii), (e)-(iv)

Step 1: Identify Equation (a) Form

Looking at equation (a) \(A x+B y+C=0\), we can see that it's in the general form of a linear equation, as it's not specifically arranged to highlight any particular features, such as slope or intercepts. So, equation (a) matches with (iii) general form.

Step 2: Identify Equation (b) Form

Equation (b) \(x=a\) represents a vertical line because for any value of \(y\), \(x\) remains constant. So, equation (b) matches with (i) vertical line.

Step 3: Identify Equation (c) Form

Equation (c) \(y=b\) represents a horizontal line because for any value of \(x\), \(y\) is fixed. So, equation (c) matches with (v) horizontal line.

Step 4: Identify Equation (d) Form

Equation (d) \(y=m x+b\) is in the slope-intercept form. It emphasizes the slope and y-intercept of the line. So, equation (d) matches with (ii) slope-intercept form.

Step 5: Identify Equation (e) Form

Equation (e) \(y-y_{1}=m\left(x-x_{1}\right)\) is the point-slope form. It's derived from the slope formula, and it opens with a given point on the line and the slope of the line. So, equation (e) matches with (iv) point-slope form.

Key Concepts

General Form

The general form of a linear equation is represented as \(Ax + By + C = 0\). This form encompasses all linear equations, regardless of their slope or intercept values. It is a versatile format because it is not specific to any particular properties of the line, like its slope or where it crosses the axes.

• It can easily be rearranged to find other forms, such as slope-intercept.
• Primarily used when you want a standard form for checking if two lines are parallel or perpendicular.
• The coefficients \(A\), \(B\), and \(C\) can be any real numbers, but \(A\) and \(B\) should not both be zero.

This equation form is especially useful in algebraic calculations because it consolidates the equation in a standard format, which aids in performing various mathematical operations.

Vertical Line

A vertical line is represented by the equation \(x = a\), where \(a\) is a constant. This means, irrespective of any changes to the \(y\)-value (up or down), the \(x\)-value remains static.

• A key feature of vertical lines is that they do not have a slope, often considered as an 'undefined' slope because they go straight up and down.
• Vertical lines are used to represent positions that do not change horizontally.
• This form is helpful in graphing situations where you need to plot lines that are parallel to the \(y\)-axis.

In geometry, vertical line equations are used often to highlight boundaries that don't shift horizontally, providing a clear demarcation.

Horizontal Line

Horizontal lines are modeled by the equation \(y = b\), with \(b\) being a constant. This signifies that the line runs parallel to the \(x\)-axis and the \(y\)-value remains the same throughout its length.

• The slope of a horizontal line is zero, since there is no vertical change as it extends.
• Horizontal lines are perfect for indicating a set dose of data points, such as constant costs or fixed values across a span.
• These lines are often utilized in graphs to demonstrate areas or levels that are consistent across various domains.

Understanding this form is crucial as it clearly demarcates a range where there’s no variation in the \(y\)-coordinate, making it a reliable reference in both graphs and equations.

Slope-Intercept Form

The slope-intercept form is given by the formula \(y = mx + b\), where \(m\) is the slope of the line, and \(b\) is the y-intercept. This form is particularly useful for graphing lines and quickly identifying key attributes of the line.

• It displays the steepness and direction of a line through its slope, \(m\).
• The y-intercept, \(b\), is where the line crosses the y-axis, and it allows easy graph plotting.
• This form is popular for solving equations because of its straightforwardness in showing how changes in \(x\) affect \(y\).

Using the slope-intercept form makes it easy to visualize the line on a graph and understand its relationship with both axes, providing immediate insights into the characteristics of the linear function.

Point-Slope Form

The point-slope form of a linear equation is expressed as \(y - y_1 = m(x - x_1)\). This form emphasizes knowing a particular point \((x_1, y_1)\) on the line and the slope \(m\).

• It’s particularly useful when you have a specific point and slope, allowing you to rapidly write the equation of the line.
• Easy to convert to other line forms like slope-intercept or general form, depending on your requirements.
• This form is helpful for understanding how a general point fits within the line's structure when the slope is known.

With point-slope form, you can derive different equation forms and quickly plot or analyze the line based on a fixed point with an understood direction, greatly aiding in line analysis and calculations.