Problem 35 - Algebra 1

Question

Write an equation in standard form of the line that passes through the given point and has the given slope. $$(-1,4), m=-3$$

Step-by-Step Solution

Answer

Therefore, the equation of the line that passes through (-1,4) and has a slope of -3 in standard form is \(3x + y = 1\).

Step 1: Put given point and slope into point-slope form

The line passes through the point (-1,4) and has a slope of -3, hence into the point-slope form of a line, values will be substituted to get an equation that represents this line. By doing so, we get: \[y - 4 = -3(x + 1).\]

Step 2: Simplify the Equation

Simplify this equation by distributing \(m\) on the right side and rearranging the equation, giving us \( y - 4 = -3x - 3.\] We then add 3x + 4 to both sides to get the equation in standard form, resulting in \(3x + y = 1.\)

Step 3: Verify Standard Form

So, \(3x + y = 1\) is the equation of the line in standard form, where A, B, and C are integers, and A \(\geq 0\), so it meets all requirements for standard form equations.

Key Concepts

Point-Slope Form

Understanding the point-slope form of a line is crucial when dealing with linear equations, especially when you have a specific point and the slope of a line. The general formula for point-slope form is expressed as \( y - y_1 = m(x - x_1) \), where \( m \) represents the slope of the line and \( (x_1, y_1) \) is the point through which the line passes.

When given a point, say \( (-1,4) \), and a slope, \( m=-3 \), we substitute these into the formula to get \( y - 4 = -3(x + 1) \). Understanding point-slope form is fundamental as it directly shows how \( y \) changes with \( x \) relative to a known point on the line, setting up the foundation for us to rearrange this into other forms of a linear equation.

Linear Equations

Linear equations form the basis of a wide range of problems and applications in mathematics. These equations describe a straight line in the coordinate system and can be presented in various formats, including standard form, slope-intercept form, and point-slope form.

The general expression for a linear equation in standard form is \( Ax + By = C \), with \( A \) and \( B \) being coefficients and \( C \) a constant. One essential characteristic of standard form is that \( A \) should be a non-negative integer. For example, after simplifying the point-slope form of our line \( y - 4 = -3(x + 1) \), we converted it to standard form \( 3x + y = 1 \). It's valuable for students to understand how to manipulate linear equations to switch between these forms, as certain forms may be more suitable in different scenarios.

Slope of a Line

The slope is a measure of how steep a line is and its direction. Mathematically, it is the ratio of the vertical change, or rise, to the horizontal change, or run, between two distinct points on the line. It's denoted by \( m \).

If you know the slope of a line and a point through which the line passes, you can write the equation of the line. For instance, with a slope of \( m = -3 \) and a point \( (-1,4) \) as in our exercise, the point-slope form made it straightforward to incorporate these values. The negative slope indicates that the line is falling; for every step we go to the right (\