Find the slope of the line given by the equation 4x + 6y = 9.
Finding the Slope of a Linear Equation
The question asks us to find the slope of the line represented by the equation \(4x + 6y = 9\).
To find the slope of a linear equation, we can convert the equation into the slope-intercept form, which is \(y = mx + c\). In this form, \(m\) represents the slope of the line, and \(c\) represents the y-intercept.
Steps to Find the Slope
We start with the given equation:
\(4x + 6y = 9\)
Our goal is to isolate the \(y\) term on one side of the equation.
- Subtract \(4x\) from both sides of the equation: \(6y = 9 - 4x\) \(6y = -4x + 9\)
- Divide both sides of the equation by the coefficient of \(y\), which is 6: \(\frac{6y}{6} = \frac{-4x + 9}{6}\) \(y = \frac{-4x}{6} + \frac{9}{6}\)
- Simplify the fractions: The fraction \(\frac{-4}{6}\) simplifies to \(\frac{-2}{3}\). The fraction \(\frac{9}{6}\) simplifies to \(\frac{3}{2}\).
So, the equation in slope-intercept form is:
\(y = \frac{-2}{3}x + \frac{3}{2}\)
Identifying the Slope
Comparing this equation to the slope-intercept form \(y = mx + c\):
\(y = \left(\frac{-2}{3}\right)x + \left(\frac{3}{2}\right)\)
Here, the coefficient of \(x\) is \(m\), which is the slope. In our equation, the coefficient of \(x\) is \(\frac{-2}{3}\).
Therefore, the slope of the line \(4x + 6y = 9\) is \(\frac{-2}{3}\).
Comparing with Options
Let's compare our calculated slope with the given options:
- Option 1: \(\frac{-3}{2}\)
- Option 2: 2/3
- Option 3: \(\frac{-2}{3}\)
- Option 4: 3/2
Our calculated slope, \(\frac{-2}{3}\), matches Option 3.
Revision Table: Forms of Linear Equations
| Form Name | Equation | Description |
|---|---|---|
| Slope-Intercept Form | \(y = mx + c\) | \(m\) is the slope, \(c\) is the y-intercept. Useful for graphing. |
| Standard Form | \(Ax + By = C\) | \(A\), \(B\), and \(C\) are constants. \(A\) and \(B\) are not both zero. Can be converted to slope-intercept form to find slope. |
Additional Information on Slope
The slope of a line is a measure of its steepness. It describes how much the line rises or falls vertically for every unit it moves horizontally. A positive slope indicates that the line goes upwards from left to right, while a negative slope indicates that the line goes downwards from left to right.
For an equation in the standard form \(Ax + By = C\), the slope (\(m\)) can be directly calculated using the formula \(m = -\frac{A}{B}\), provided \(B \neq 0\). Let's check this with our equation \(4x + 6y = 9\).
Here, \(A=4\), \(B=6\), and \(C=9\).
Using the formula \(m = -\frac{A}{B}\):
\(m = -\frac{4}{6}\)
\(m = -\frac{2}{3}\)
This confirms the slope we found by converting to slope-intercept form. This direct formula can be a quick way to find the slope from the standard form.
Similar Questions
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Important Questions from Co-ordinate Geometry
The graphs of the linear equations 4x - 2y = 10 and 4x + ky = 2 intersect at a point (a, 4). The value of k is equal to:
In which ratio the point (-3, p) divides the line segment joining the points (-5, -4) and (-2, 3)?
The area (in sq. units) of the triangle formed by the graphs of 8x + 3y = 24, 2x + 8 = y and the x-axis is:
In which quadrant both abscissa and ordinate are negative?
Find the slope of the line joining the points (3, -4) and (5, 2).
