Problem 7 - 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. Use integer coefficients. \((1,-2), m=5\)

Step-by-Step Solution

Answer

The equation of the line in standard form is \(-5x + y = -7\).

Step 1: Identify the given values

From the exercise, the point given is \((1, -2)\), so \(x_1 = 1\) and \(y_1 = -2\). The slope 'm' is given as 5.

Step 2: Write the equation in point-slope form

We know the point-slope form is given by \(y - y_1 = m(x - x_1)\). Plugging in the values for \(x_1\), \(y_1\), and \(m\), we obtain \(y - (-2) = 5(x - 1)\) which simplifies to \(y + 2 = 5x - 5\).

Step 3: Convert the equation from point-slope form to standard form

The equation from step 2 can be re-arranged into the standard form. Subtract \(5x\) from both sides and subtract 2 from both sides, the equation becomes: -5x + y = -7.

Key Concepts

Point-Slope Form

When you hear about the point-slope form of a line, think of it as a simple recipe for baking up equations. It is an equation format that makes it easy to create the line equation when you know a specific point on the line and the slope. The mathematical form is written as \[y - y_1 = m(x - x_1)\] Here’s what each part means:

• \(y\) and \(x\) stand for the variables we'll be solving for.
• \(y_1\) and \(x_1\) are the coordinates of the specific point on the line. They act as the anchor of your line.
• \(m\) is the slope, describing how steep your line is.

To use the point-slope form, just plug in the \((x_1, y_1)\) and slope \(m\) given in the problem.For example, with point \((1, -2)\) and slope \(5\), you get: \[y - (-2) = 5(x - 1)\] This simplifies to: \[y + 2 = 5x - 5\]This form is particularly great because it allows you to quickly draft the equation of a line even when you only have limited information.

Slope

The slope of a line is like the backbone of a line equation. It tells you how fast or steep the line rises or falls, dependent on how much you move along the \(x\)-axis. Think of it as the tilt of the line.In mathematical terms, the slope \(m\) is calculated as:\[m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}\]This means, the slope measures the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. In our original problem, the slope is given as \(5\), which indicates that for every single unit you move right on the \(x\)-axis, the line moves up 5 units.Understanding slope helps you figure out:

• Whether a line is increasing or decreasing
• How dramatically the line angles towards the \(y\)-axis
• Rates of change in real-world contexts

Overall, the slope is a fundamental part of describing and understanding linear relationships.

Integer Coefficients

Integer coefficients make equations cleaner and simpler, especially when converting to standard form. Standard form equations like \(Ax + By = C\) require \(A\), \(B\), and \(C\) to be integers.To convert an equation with decimal or fractional coefficients to integer coefficients, you might need to multiply the entire equation by a common factor to get rid of fractions or decimals.In our exercise, the standard form we aim for is \(-5x + y = -7\), and here,

• \(A = -5\)
• \(B = 1\)
• \(C = -7\)

These are all clean, uncomplicated integers.Why is this important? Using integer coefficients in equations allows for easier interpretation and manipulation of the equations, especially in educational settings. This is because fractions and decimals can sometimes complicate the calculations and understanding of the problem.By converting equations to have integer coefficients, we ensure they are straightforward to solve and analyze.