Problem 26 - Algebra 1: Concepts and Skills

Question

Write in standard form an equation of the line that passes through the given point and has the given slope. \((10,6), m=0\)

Step-by-Step Solution

Answer

The equation of the line in standard form that passes through the point \((10,6)\) and has the slope \(m = 0\) is \(0x + y = 6\).

Step 1: Calculate the y-intercept \(c\)

Since the line passes through the point \((10,6)\), and has slope \(m=0\), the y-intercept can be found using the formula \(c = y - mx\), thus \(c = 6 - 0*10 = 6\). Now the equation of the line can be written as \(y = 0x + 6\).

Step 2: Convert to standard form

Rewrite the equation in standard form 'Ax + By = C'. The standard form implies that A, B, and C should be integers, B should be positive, and A and B should not both be zero. Additionally, A, B, and C have no common factors except 1. Start by multiplying each side of the equation \(y = 0x + 6\) by 1 (to avoid changing the equation), getting \(y = 6\). Then we rearrange the equation to get it in the standard form, hence 0x - y = -6.

Step 3: Rewrite the standard form

Since the standard form requires that \(A\), \(B\) and \(C\) have no common factors except 1 and that \(A\) and \(B\) should not both be zero, we multiply equation 0x - y = -6 by -1 to get 0x + y = 6, where \(A = 0\), \(B = 1\) and \(C = 6\).

Key Concepts

Standard Form of a Line

The standard form of a line is an alternative way to write a linear equation and is commonly represented as \( Ax + By = C \). In this notation, \( A \), \( B \) and \( C \) are integers where \( A \) should be nonnegative, and \( A \) and \( B \) should not both be zero. When converting an equation to standard form, if \( B \) is negative, you can multiply the entire equation by -1 to ensure \( B \) is positive.

When faced with the task of writing an equation in standard form, start by ensuring that the coefficients for \( x \) and \( y \) are integers without any common factors. Next, ensure that the \( x \) term is written first, followed by the \( y \) term, and lastly the constant term, like in the given exercise for step 2. The example in the exercise illustrates that even if the \( x \) coefficient is zero, it meets the criteria since \( A \) and \( B \) are not both zero. We then ended up with the equation in the format desired: \( 0x + y = 6 \) as the final standard form.

Algebraic Slope-Intercept Form

The slope-intercept form is one of the most commonly used expressions for a straight line in algebra. It is written as \( y = mx + c \), where \( m \) represents the slope of the line, and \( c \) is the y-intercept, which is the point where the line crosses the y-axis. This form is convenient as it explicitly showcases the slope and y-intercept, making it easy to graph.

In the case of the exercise, having the slope, \( m \), equal to zero indicates that the line is horizontal, as any value of \( x \) will not affect the value of \( y \) which stays constant. Therefore, for the slope-intercept form \( y = 0x + 6 \) the value \( 6 \) gives away the y-intercept without additional calculation. It also reinforces the concept that a horizontal line's equation can be simplified to \( y = c \), which is a special form of the slope-intercept equation.

Y-Intercept Calculation

The y-intercept of a line is the point where it crosses the y-axis and can be determined algebraically from an equation or graphically from a plot. Algebraically, the y-intercept is found by setting \( x \) to zero and solving for \( y \). In our exercise, since the slope (\( m \) ) is zero, the y-intercept (\( c \) ) is simply the \( y \) value of the given point through which the line passes (\( 10,6 \) ), resulting in \( c = 6 \).

The step-by-step solution confirms this by utilizing the formula \( c = y - mx \), yielding \( c = 6 - 0*10 = 6 \), providing students with an indispensable method to find the y-intercept from a slope and a point on the line. The calculated y-intercept can then be easily plugged into the slope-intercept form or converted into standard form, depending on the requirements of the problem.