Problem 10 - Elementary Algebra

Question

For Problems 1-12, 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. $$ (0,0), m=\frac{5}{11} $$

Step-by-Step Solution

Answer

The equation of the line is \(5x - 11y = 0\).

Step 1: Understand Point-Slope Formula

To find the equation of the line, we start with the point-slope form of the equation, which is given by \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is a point on the line.

Step 2: Substitute Given Values

Substitute the point \((0, 0)\) and the slope \(m = \frac{5}{11}\) into the point-slope form: \(y - 0 = \frac{5}{11}(x - 0)\). This simplifies to \(y = \frac{5}{11}x\).

Step 3: Eliminate Fractions

To express the equation in the form \(Ax + By = C\), we eliminate the fraction by multiplying through by 11: \(11y = 5x\).

Step 4: Rearrange to Standard Form

Rearrange the equation to get it in the desired form \(Ax + By = C\). Move all terms to one side: \(5x - 11y = 0\).

Key Concepts

Point-Slope Form

The point-slope form is a versatile method for writing the equation of a line. It comes handy when you are equipped with a point on the line and the slope of the line. The formula for the point-slope form looks like this: \( y - y_1 = m(x - x_1) \). In this formula:

• \( (x_1, y_1) \) represents a specific point on the line.
• \( m \) is the slope of the line.

Using this method, you substitute the values you know, including your point and slope, directly into the formula. This approach provides a quick path to the equation of a line. If you know that the slope \( m \) is \( \frac{5}{11} \) and your point is \((0,0)\), your equation begins as \(y - 0 = \frac{5}{11}(x - 0)\). The nice thing about this formula is its simplicity, but you'll often want to translate it into other forms, like the slope-intercept or standard form, for broader applications.

Slope-Intercept Form

The slope-intercept form is probably the most commonly used equation of a line form because of its intuitive understanding. It looks like this: \(y = mx + b\). This form explicitly shows:

• The slope \(m\), which tells you how steep the line is.
• The y-intercept \(b\), which indicates where the line crosses the y-axis.

In many exercises, converting the equation of a line to this form makes it easier to graph and analyze. For instance, after using the point-slope method, we found an equation \(y = \frac{5}{11}x\). From this result, we can easily see the slope is \(\frac{5}{11}\) and the y-intercept is 0, which implies that the line passes through the origin.

Standard Form of a Line

The standard form of a line's equation is \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers, and \(A\) should preferably be non-negative. This form is particularly useful for certain algebraic techniques and for clearly understanding relationships in linear programming and systems of equations.

To reach the standard form from the slope-intercept form \(y = \frac{5}{11}x\), we first eliminate any fractions by multiplying the entire equation by 11. This gives us \(11y = 5x\). Rearranging terms, we arrive at \(5x - 11y = 0\). The standard form tidy ups the equation into integer values, which is often required for short answer questions or standardized tests. Remember, although each form might present the equation differently, they all represent the same line.