Question

If the lines \(\mathrm{x}+\mathrm{y}+\mathrm{r}=0\) and \(\lambda \mathrm{x}-5 \mathrm{y}=5\) are identical then \(\lambda+\mathrm{r}=\ldots \ldots\) (a) \(-4\) (b) 4 (c) 1 (d) \(-1\)

Short answer

The lines \(x+y+r=0\) and \(\lambda x - 5y = 5\) are identical, which means they have the same slope and y-intercept. The slope of the first line is -1, and the y-intercept is -r. The slope of the second line is \(\frac{\lambda}{5}\), and the y-intercept is -1. Equating the slopes and y-intercepts, we have \(-1 = \frac{\lambda}{5}\) and \(-r = -1\). Solving these equations, we get \(\lambda = -5\) and \(r = 1\). Therefore, the sum \(\lambda + r = -5 + 1 = -4\), which is the correct answer (a).

Step-by-step solution

Find the slopes and y-intercepts of the given lines

To find the slope and y-intercept of a given line, rewrite the equation in the form \(y = mx + b\), where m is the slope and b is the y-intercept. For the first line, \(x+y+r=0\), we can rewrite it as: \(y = -x-r\) The slope (m1) of this line is -1, and the y-intercept (b1) is -r. For the second line, \(\lambda x - 5y = 5\), rewrite it as: \(y = \frac{\lambda}{5} x - 1\) The slope (m2) of this line is \(\frac{\lambda}{5}\), and the y-intercept (b2) is -1.

Set up equations for identical lines

Since the lines are identical, their slopes and y-intercepts must be equal. We can set up two equations based on this information: \(m_1 = m_2\) \(-1 = \frac{\lambda}{5}\) and \(b_1 = b_2\) \(-r = -1\)

Solve the system of equations

Solving for r in \(b_1 = b_2\): \(-r = -1\) Multiplying both sides by -1: \(r=1\) Now, solving for \(\lambda\) in \(m_1 = m_2\): \(-1 = \frac{\lambda}{5}\) Multiplying both sides by 5: \(-5 = \lambda\)

Find the sum of \(\lambda\) and r

Add the values of \(\lambda\) and r: \(\lambda + r = -5 + 1 = -4\) Hence, the correct answer is (a) \(-4\).

Key concepts

Slopes of lines

The slope of a line in coordinate geometry represents how steep a line is and its direction. Essentially, it describes the rate at which the 'y' value rises or falls as the 'x' value increases.

In a linear equation in the form of y = mx + b, the coefficient m stands for the slope. If two lines are identical, their slopes must be exactly the same. To evaluate slopes algebraically, you may have to manipulate the original equation into this slope-intercept form. For example, with the equation x + y + r = 0, by rearranging this to y = -x - r, we clearly see that the slope is -1.

Understanding and identifying slopes is fundamental in determining the relationship between two lines - whether they're identical, parallel, or intersecting at an angle.

Y-intercept

The y-intercept of a line is the point where the line crosses the y-axis. In the equation of a line y = mx + b, the term b represents the y-intercept. Knowing a line's y-intercept is crucial when graphing it or comparing it to another line.

For the given lines to be identical, not only their slopes but also their y-intercepts must match. The y-intercept is easily identified once you manipulate the equation into the slope-intercept form. In the example of \(\lambda x - 5y = 5\), after rearranging it to y = \(\frac{\lambda}{5}\)x - 1, it becomes apparent that the y-intercept is -1. This characteristic helps us solve for the constants when setting two lines to be identical.

System of Equations

A system of equations is a collection of two or more equations with the same set of variables. In coordinate geometry, when determining if two lines are identical, we create a system of equations based on their slopes and y-intercepts.

In our example, the system includes the equations m1 = m2 and b1 = b2. Solving this system involves finding a common solution that satisfies all equations. There are several methods to solve a system of equations, including substitution, elimination, and graphical methods. When the system derives from identical lines, the solution will provide the values of variables that make both equations represent the same line.

Algebraic Manipulation

Algebraic manipulation involves rearranging, combining, and simplifying expressions and equations to solve for variables or analyze relationships between them. This skill is extremely useful in coordinate geometry, especially when working with linear equations.

For example, in the steps to determine the relationship between r and \(\lambda\), we used algebraic manipulation to isolate variables and solve the respective equations. Initially, we rearranged the given equations into the slope-intercept form to identify the slopes and y-intercepts. Later, we manipulated these equations to solve the system, finding r and \(\lambda\). The ability to manipulate algebraic expressions thus becomes the underpinning of solving geometry problems.