Problem 9 - Elementary Algebra

Question

Find the equation of the line that contains the given point and has the given slope. Express equations in the form \(A x+B y=C\), where \(A, B\), and \(C\) are integers. (Objective 1a) $$(0,0), m=-\frac{4}{9}$$

Step-by-Step Solution

Answer

The line equation is \( 4x + 9y = 0 \).

Step 1: Understand the Point-Slope Form

The point-slope form of the equation of a line is \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) is a point on the line and \(m\) is the slope. Given point is \( (0, 0) \) and slope \( m = -\frac{4}{9} \).

Step 2: Substitute into Point-Slope Form

Substitute \( (0,0) \) for \((x_1, y_1)\) and \(-\frac{4}{9}\) for \(m\) into the point-slope form equation: \[ y - 0 = -\frac{4}{9}(x - 0) \] This simplifies to: \[ y = -\frac{4}{9}x \]

Step 3: Transform to Standard Form

The standard form of a line is \( Ax + By = C \). We need to eliminate the fraction and rewrite the equation in this form. Multiply through by 9 to clear the fraction:\[ 9y = -4x \] Rearrange terms to fit the form \( Ax + By = C \): \[ 4x + 9y = 0 \]

Step 4: Verify Integer Coefficients

Check if \(A, B,\) and \(C\) are integers. Here, \(A = 4\), \(B = 9\), and \(C = 0\) are all integers, thus satisfying the requirement.

Key Concepts

Point-Slope Form

The point-slope form is a handy way to create the equation of a line when you know a point on the line and its slope. The formula is expressed as \( y - y_1 = m(x - x_1) \). Here, \((x_1, y_1)\) are the coordinates of the point, and \(m\) is the slope.
Using this form helps since it's direct and simple; you only need two pieces of information: one point on the line and the slope. Once you have these, you can quickly plug them into the formula. This form is particularly helpful for forming linear equations when initially starting with known values.
In this case, given the point \((0, 0)\) and the slope \(-\frac{4}{9}\), you substitute directly into the formula and simplify to get:

• Initial Step: \( y - 0 = -\frac{4}{9}(x - 0) \)
• Simplified: \( y = -\frac{4}{9}x \)

The result \( y = -\frac{4}{9}x \) is the equation of the line in point-slope form before converting into another form.

Standard Form

The standard form of a linear equation is presented as \( Ax + By = C \). This format is useful for expressing equations because it highlights the coefficients of \(x\) and \(y\), and it allows for a more straightforward exposition of the line in geometry.
To convert from point-slope form to standard form, some manipulation of the equation is necessary. It's common to remove fractions and adjust terms so \(x\) and \(y\) are on one side, leaving a constant on the other.
After simplifying \( y = -\frac{4}{9}x \), multiply every term by 9 to get rid of the fractional coefficient.

• Multiply: \( 9y = -4x \)
• Rearrange: \( 4x + 9y = 0 \)

The result \( 4x + 9y = 0 \) is the line's equation in standard form. This method aligns well with algebraic conventions and can facilitate further analysis of the line's geometric properties.

Integer Coefficients

Integer coefficients in equations ensure the numbers used in the equations are whole numbers, which simplifies calculations and understanding.
When writing linear equations, especially in standard form, it's important that coefficients \(A\), \(B\), and \(C\) are integers. This requirement keeps the equations neat and avoids unnecessary complications with fractions or decimals.
By expressing \( 4x + 9y = 0 \) with integer coefficients:

• \(A = 4\)
• \(B = 9\)
• \(C = 0\)

All these values are integers, meeting the standards for an equation that's easily interpretable and usable in practical contexts without additional conversion. This characteristic is crucial for effective communication in mathematics.