Problem 62 - Algebra 1: Concepts and Skills

Question

Choose an equation in standard form of the line that passes through the point \((-1,-4)\) and has a slope of \(2 .\) (F) \(-2 x+y=-2\) (H) \(-x-y=9\) (G) \(-3 y=x-9\) (J) \(x-3 y=-9\)

Step-by-Step Solution

Answer

The correct equation of the line that passes through the point (-1,-4) and has a slope of 2 in standard form is \(-2x + y = -2\). So, the correct choice is (F)

Step 1 : Writing the Equation in Point-Slope Form

Firstly, the equation will be written in point-slope form. The form is defined as \(y - y_1 = m(x - x_1)\) where \(m = 2\) (given slope), \((x_1 = -1, y_1 = -4)\). By plugging these values, we will have \(y - (-4) = 2(x - (-1))\).

Step 2 : Simplifying The Equation

Simplify the equation by resolving the negative signs. They become \(y + 4 = 2(x + 1)\). Upon further simplification, we get \(y + 4 = 2x + 2\).

Step 3 : Transforming to Standard form

Finally, the equation is written in standard form, which usually is designed as \(Ax + By = C\). By subtracting \(2x\) from both sides, the equation becomes \(-2x + y = -2\).

Key Concepts

Standard Form

Standard form is a way of writing the equation of a line such that it appears as: \[Ax + By = C\]Here, \(A\), \(B\), and \(C\) are integers, and the convention is to have \(A\) as a positive number, if possible. It's particularly useful because it allows us to easily identify the x-intercept and y-intercept of a line and works well with inequalities.

• One common way to convert other forms of line equations into standard form is by rearranging terms and simplifying.
• The coefficients \(A\), \(B\), and \(C\) should ideally be integers, and any fractions should be cleared by multiplying through by the denominator.
• In the given solution, the line equation \(-2x + y = -2\) is the outcome of transforming from the point-slope form into standard form, aligning with the standard format.

Standard form is often preferred when dealing with systems of equations as it allows for easier manipulation and combination of equations.

Point-Slope Form

The point-slope form is a straightforward way of writing the equation of a line when you know one point on the line and the slope. The formula is given as:\[y - y_1 = m(x - x_1)\]where \((x_1, y_1)\) is a known point on the line, and \(m\) is the slope.This form is extremely helpful as it directly uses the most basic elements needed to plot a line: a point and the direction in which the line goes.

• It makes it easier to write the equation of a line, as seen in the original exercise, where \((-1,-4)\) was used along with the slope \(2\).
• Once the equation is set up with the point-slope form, you can convert it into other forms, such as slope-intercept or standard form, by following simple algebraic steps.
• The initial equation \(y + 4 = 2(x + 1)\) began as a point-slope form before being converted in the problem's later steps.

This form serves as a versatile starting point for deriving other equations of the same line.

Slope-Intercept Form

The slope-intercept form is frequently used for graphing linear equations because of its simplicity. It is presented as:\[y = mx + b\]where \(m\) represents the slope of the line and \(b\) is the y-intercept, the point where the line crosses the y-axis.The clear advantage of this format is that it allows you to see at a glance how steep the line is and where it intersects the y-axis.

• It's especially handy in graphing tasks because you can start plotting from the y-intercept and use the slope to find other points.
• From the exercise's transformed equation \(y + 4 = 2x + 2\), isolating \(y\) leads directly to a slope-intercept form: \(y = 2x - 2\).
• The straightforwardness of this form makes it ideal for beginners in algebra who are learning to graph solutions on a coordinate plane.

Utilizing slope-intercept form offers an efficient method for visualizing and understanding the behavior of linear equations.