Problem 33 - Algebra 1: Concepts and Skills

Question

Write in standard form an equation of the line that passes through the two points. Use integer coefficients. \((0,1),(1,-1)\)

Step-by-Step Solution

Answer

The equation of the line that passes through the points (0,1) and (1,-1) in standard form is \(2x + y = 1\).

Step 1: Find The Slope

The formula for the slope is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). By substituting \( x_1=0, y_1=1, x_2=1, y_2=-1 \) into the formula you get \( m = \frac{-1 - 1}{1 - 0} = -2 \). The slope of the line is -2.

Step 2: Finding the Y-Intercept

We can use either point to find the y-intercept (b), but we'll use point (0,1) because it is easier. Substitute m = -2, x = 0, and y = 1 into the formula \( b = y - mx \) to get \( b = 1 - (-2)*0 = 1 \). The y-intercept of the line is 1.

Step 3: Writing the Equation

Substitute m = -2 and b = 1 into the equation \( y = mx + b \) You get \( y = -2x + 1 \). However, we need to write the equation in standard form (Ax + By = C). Standard form is when A (the x coefficient) is positive and A, B and C are integers and A and B are not both zero, so rearrange the equation to get \(2x + y = 1\)

Key Concepts

Slope Calculation

When dealing with linear equations, one of the first steps is to calculate the slope of the line. The slope shows how steep the line is and the direction it goes. It’s found using the formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Here, \(x_1\), \(y_1\) are the coordinates of the first point and \(x_2\), \(y_2\) of the second point. Think of the slope as the rise over the run; it tells us how much \(y\) changes for a change in \(x\).
For example, with the points \((0,1)\) and \((1,-1)\), substituting into the formula gives:

• \(x_1 = 0\), \(y_1 = 1\)
• \(x_2 = 1\), \(y_2 = -1\)

Replacing in the formula, we calculate:\[ m = \frac{-1 - 1}{1 - 0} = -2 \]This tells us the slope \(m\) of the line is \(-2\), indicating the line decreases as \(x\) increases.

Standard Form Equation

The standard form of a linear equation is written as:\[ Ax + By = C \]Where \(A\), \(B\), and \(C\) are integers, and \(A\) should be positive. This format is handy when it comes to graphing or finding intercepts because it presents everything neatly on one side. To convert from the slope-intercept form \(y = mx + b\) to standard form, rearrange the equation to place \(x\) and \(y\) on one side.
Using the equation we calculated, \(y = -2x + 1\), we rearrange like this:

• Add \(2x\) to both sides: \(y + 2x = 1\)
• Reorder terms: \(2x + y = 1\)

This gives our line equation in standard form, where \(A = 2\), \(B = 1\), and \(C = 1\), all of which meet the integer requirements.

Y-Intercept

The y-intercept is the point where the line crosses the y-axis. This is the value of \(y\) when \(x\) equals zero. To find the y-intercept, we use the equation:\[ b = y - mx \]Choosing an easy point like \((0,1)\) simplifies calculations because \(x=0\) on the line. Plugging in the values:

• \(m = -2\)
• \(x = 0\)
• \(y = 1\)

We substitute into the equation for \(b\):\[ b = 1 - (-2) \cdot 0 = 1 \]Thus, the y-intercept \(b\) is \(1\). This shows the point where the line meets the y-axis, giving us an essential anchor point for the line on a graph.