Representation of a Line-2

Question 1: Equation from a Point and Slope

The Question: The equation of a line passing through $(-1, -1)$ with value of slope 1 is __________.

• $y = -x$
• $y = x$
• $y = -x - 1$
• $y = -x + 1$

Core Concept: The Point-Slope Form

When you are given a point $(x_1, y_1)$ on a line and its slope ($m$), the most direct way to find the equation is using the point-slope form:

You can then simplify this equation into the more common slope-intercept form ($y = mx + c$).

Detailed Solution:

1. Identify the given information:

   • The point is $(x_1, y_1) = (-1, -1)$.
   • The slope is $m = 1$.

2. Substitute these values into the point-slope formula:

   $$y - (-1) = 1 \times (x - (-1))$$

3. Simplify the equation:

   $$y + 1 = 1(x + 1)$$
   $$y + 1 = x + 1$$

   Subtract 1 from both sides:

   $$y = x$$

Final Answer: The equation of the line is $y = x$.

Question 2: Equation from Two Intercepts

The Question: The equation of a line which cuts the X-axis at $(5, 0)$ and Y-axis at $(0, 5)$ is __________.

• $x + y = -1$
• $y = -x + 5$
• $-y = -x + 1$
• $y = 5x + 5$

Core Concept: The Intercept Form

When a line’s x-intercept (where it crosses the x-axis, at point $(a, 0)$) and y-intercept (where it crosses the y-axis, at point $(0, b)$) are known, you can use the intercept form:

Detailed Solution:

We can solve this in two ways.

Method 1: Using the Intercept Form

1. Identify the intercepts:
   • The x-intercept is at $(5, 0)$, so $a = 5$.
   • The y-intercept is at $(0, 5)$, so $b = 5$.
2. Substitute into the intercept formula: $$\frac{x}{5} + \frac{y}{5} = 1$$
3. Simplify the equation:
   • Multiply the entire equation by 5 to eliminate the denominators:
   • $x + y = 5$
   • To match the options, isolate y:
   • $y = -x + 5$

Method 2: Using the Two-Point Form

1. Find the slope ($m$) using the points $(5, 0)$ and $(0, 5)$:
   • $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - 0}{0 - 5} = \frac{5}{-5} = -1$.
2. Use the slope-intercept form ($y = mx + c$):
   • We know the slope $m = -1$.
   • The y-intercept is given as the point $(0, 5)$, so $c = 5$.
   • Substitute the values: $y = (-1)x + 5$, which simplifies to $y = -x + 5$.

Final Answer: The equation of the line is $y = -x + 5$.

Question 3: Finding the X-Intercept

The Question: The X-intercept of a line $y = 3x + 1$ is __________.

• -1
• 1/2
• -1/2
• -1/3

Core Concept: Finding Intercepts

• X-Intercept: The point where a line crosses the x-axis. At this point, the y-coordinate is always 0. To find the x-intercept, set $y=0$ in the equation and solve for $x$.
• Y-Intercept: The point where a line crosses the y-axis. At this point, the x-coordinate is always 0.

Detailed Solution:

1. Start with the equation:
   • $y = 3x + 1$
2. Set y = 0 to find the x-intercept:
   • $0 = 3x + 1$
3. Solve the equation for x:
   • Subtract 1 from both sides: $-1 = 3x$
   • Divide by 3: $x = -\frac{1}{3}$
   • The x-intercept is the point $(-1/3, 0)$. The question asks for the x-intercept value.

Final Answer: The X-intercept is -1/3.

Question 4: Analyzing Linear Equations

The Question: Choose the correct answers. (Multiple Select Question)

• $y = 2x - 8$ is the equation of a line in slope intercept form where -8 is Y-intercept.
• $y = 0$ is the equation of a line passing through the origin.
• $y = 2x - 8$ does not meet either the origin and the X-axis.
• $y = -2(x+1)$ is the equation of a line in intercept form.

Core Concepts: Forms of Linear Equations

• Slope-Intercept Form: $y = mx + c$, where $m$ is the slope and $c$ is the y-intercept.
• Intercept Form: $\frac{x}{a} + \frac{y}{b} = 1$, where $a$ is the x-intercept and $b$ is the y-intercept.
• Passing through the Origin: A line passes through the origin $(0, 0)$ if substituting $x=0$ and $y=0$ into the equation results in a true statement. For $y = mx + c$, this means $c=0$.

Detailed Solution:

Let’s evaluate each statement:

1. "$y = 2x - 8$ is the equation of a line in slope intercept form where -8 is Y-intercept."

   • The equation is in the form $y = mx + c$, with $m=2$ and $c=-8$.
   • This is the definition of the slope-intercept form, and the y-intercept is indeed -8. This statement is TRUE.

2. "$y = 0$ is the equation of a line passing through the origin."

   • Let’s check if the point $(0, 0)$ satisfies the equation. If we substitute $x=0$ and $y=0$, we get $0=0$, which is a true statement.
   • Therefore, the line $y=0$ (which is the x-axis) passes through the origin. This statement is TRUE.

3. "$y = 2x - 8$ does not meet either the origin and the X-axis."

   • Let’s check the origin $(0,0)$: $0 = 2(0) - 8 \implies 0 = -8$. This is false, so it does not meet the origin.
   • Let’s check the X-axis (where $y=0$): $0 = 2x - 8 \implies 2x = 8 \implies x = 4$. The line does meet the X-axis at the point $(4,0)$.
   • Since the statement claims it meets neither, but it does meet the X-axis, the entire statement is FALSE.

4. "$y = -2(x+1)$ is the equation of a line in intercept form."

   • The standard intercept form is $\frac{x}{a} + \frac{y}{b} = 1$.
   • Let’s rewrite $y = -2(x+1)$ as $y = -2x - 2$. This is the slope-intercept form.
   • The given form is not the standard intercept form. This statement is FALSE.

Final Answer: The correct options are "$y = 2x - 8$ is the equation of a line in slope intercept form where -8 is Y-intercept" and "$y = 0$ is the equation of a line passing through the origin".