Question

Find \(\lambda\), if the line \(3 x-\lambda y+6=0\) passes through the point \((-3,4)\). (1) \(\frac{3}{4}\) (2) \(\frac{-3}{4}\) (3) \(\frac{4}{3}\) (4) \(\frac{-4}{3}\)

Short answer

Answer: (2) \(\frac{-3}{4}\)

Step-by-step solution

Substitute the point coordinates

Plug in the coordinates of the point \((-3, 4)\) into the equation of the line, replacing \(x\) with \(-3\) and \(y\) with \(4\): $$3(-3) - \lambda(4) + 6 = 0$$

Solve for \(\lambda\)

Simplify the equation and solve for \(\lambda\): $$-9 - 4\lambda + 6 = 0$$ $$-3 - 4\lambda = 0$$ Subtracting \(-3\) from both sides gives us: $$-4\lambda = 3$$ Now divide both sides by \(-4\) to isolate \(\lambda\): $$\lambda = \frac{3}{-4}$$

Check the answer

The possible answers were: (1) \(\frac{3}{4}\) (2) \(\frac{-3}{4}\) (3) \(\frac{4}{3}\) (4) \(\frac{-4}{3}\) We found that \(\lambda = \frac{3}{-4} = \frac{-3}{4}\), so the correct answer is (2) \(\frac{-3}{4}\).

Key concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, allows us to study geometric shapes using a system of coordinates in a plane. We use the Cartesian coordinate system where each point on the plane is represented by an ordered pair \((x, y)\). In the given exercise, the point \((-3, 4)\) is a crucial part of discovering the value of \(\lambda\) in the linear equation. This specific coordinate pair represents a location in the plane which the line must pass through.

• Coordinate: A pair of numbers \((x, y)\)
• Plane: A flat surface extending infinitely in two dimensions
• Line: A straight one-dimensional figure having no thickness and extending infinitely

By understanding coordinate geometry, you can determine the relationships between different geometric shapes and lines on the coordinate plane.

Point-Slope Form

The point-slope form is a handy way to form the equation of a line. It requires a known point \((x_1, y_1)\) and the slope \(m\) of the line. The generic form is given as: \[ y - y_1 = m(x - x_1) \]In the exercise, however, we use a different form, \(3x - \lambda y + 6 = 0\), which is still connected to the general principles of point-slope.

• Known point: A specific point \((x_1, y_1)\) through which the line passes
• Slope \(m\): The rate of change indicating the steepness of the line as it moves across the plane

Understanding the point-slope concept helps in determining missing parameters, like \(\lambda \), when given specific points that lie on the line.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations to solve for unknown variables. In this exercise, we need to find \(\lambda\) by substituting the given point into the line equation. This requires several algebraic steps:

• Substitute the coordinates for \(x\) and \(y\)
• Simplify the equation
• Isolate \(\lambda\) by solving the equation

For example, substituting \(-3\) for \(x\) and \(4\) for \(y\) and then simplifying yields: \(-3 - 4\lambda = 0\).
Solving for \(\lambda\) gives \(\lambda = \frac{-3}{4}\). Employing these methods ensures you extract the value of \(\lambda\) accurately.

Solution Verification

Solution verification is the final step where you confirm your answer is correct by comparing it to the given options. After computing \(\lambda=\frac{-3}{4}\), the correctness can be ascertained by matching it with the options provided:

• (1) \(\frac{3}{4}\)
• (2) \(\frac{-3}{4}\)
• (3) \(\frac{4}{3}\)
• (4) \(\frac{-4}{3}\)

The value \(\lambda=\frac{-3}{4}\) precisely matches option (2), which confirms the solution.
It’s essential to ensure that all calculations are correctly done to verify that the solution is both mathematically sound and corresponds with provided multiple-choice answers.