General equation of line

Questions 4, 5, & 6: Analyzing a Linear Equation (from file image_0b5b27.png)

The Equation: The equation of a line is $4x - 2y - 12 = 0$. Answer the following questions.

Core Concepts: General Form, Slope-Intercept Form, and Intercepts

The given equation is in the General Form, $Ax + By + C = 0$. To easily find the slope and intercepts, we can convert it.

• Slope-Intercept Form ($y = mx + c$): By isolating $y$, we can directly find the slope ($m$) and the y-intercept ($c$).
• Finding the x-intercept: This is the point where the line crosses the x-axis. At this point, the y-value is always 0. To find it, we set $y=0$ in the equation and solve for $x$.
• Finding the y-intercept: This is the point where the line crosses the y-axis. At this point, the x-value is always 0. We can find it by setting $x=0$ or by looking at the ‘$c$’ value in the slope-intercept form.

4) What is the slope of the line?

Detailed Solution:

1. Start with the general form: $4x - 2y - 12 = 0$
2. Isolate the y-term: $-2y = -4x + 12$
3. Divide the entire equation by -2 to solve for y: $y = \frac{-4x}{-2} + \frac{12}{-2}$
4. Simplify: $y = 2x - 6$
5. Identify the slope: This equation is now in the form $y = mx + c$. The slope ($m$) is the coefficient of $x$.

Final Answer: The slope of the line is 2.

5) What is the x-intercept?

Detailed Solution:

1. Start with the original equation: $4x - 2y - 12 = 0$
2. Set y = 0: $4x - 2(0) - 12 = 0$
3. Simplify and solve for x: $4x - 0 - 12 = 0$ $4x = 12$ $x = \frac{12}{4} = 3$

Final Answer: The x-intercept is 3.

6) What is the y-intercept?

Detailed Solution:

Method 1: Using the Slope-Intercept Form From our work in question 4, we converted the equation to $y = 2x - 6$. In the form $y = mx + c$, the y-intercept is the value of $c$.

Final Answer: The y-intercept is -6.

Method 2: Setting x = 0

1. Start with the original equation: $4x - 2y - 12 = 0$
2. Set x = 0: $4(0) - 2y - 12 = 0$
3. Simplify and solve for y: $0 - 2y - 12 = 0$ $-2y = 12$ $y = \frac{12}{-2} = -6$

Final Answer: The y-intercept is -6.

Question 1: Equation from Intercepts (from file image_0b62af.png)

The Question: If the x-intercept and the y-intercept of a straight line are -6 and 7 respectively, then choose the correct equation of the line.

• $7x - 6y + 42 = 0$
• $-6x + 7y - 1 = 0$
• $7x - 6y - 1 = 0$
• $-6x + 7y - 2 = 0$

Core Concept: The Intercept Form of a Line

When you are given the x-intercept ($a$) and the y-intercept ($b$), the quickest way to find the equation is the intercept form:

Detailed Solution:

1. Identify the intercepts:
   • x-intercept $a = -6$
   • y-intercept $b = 7$
2. Substitute into the intercept formula: $$\frac{x}{-6} + \frac{y}{7} = 1$$
3. Convert to the General Form ($Ax+By+C=0$) to match the options:
   • To eliminate the denominators, multiply the entire equation by the least common multiple of -6 and 7, which is -42.
   • $(-42) \times \frac{x}{-6} + (-42) \times \frac{y}{7} = (-42) \times 1$
   • $7x - 6y = -42$
4. Rearrange the equation:
   • Move all terms to one side to set the equation to 0.
   • $7x - 6y + 42 = 0$

Final Answer: The correct equation is $7x - 6y + 42 = 0$.

Question 2: Finding the Slope (from file image_0b62af.png)

The Question: The slope of the line $6x - 2y + 8 = 0$ is __________.

• 2
• 3
• 5
• 1

Core Concept: Finding Slope from the General Form

To find the slope of a line given in the general form $Ax + By + C = 0$, the most reliable method is to convert it to the slope-intercept form, $y = mx + c$, by solving for $y$. The coefficient of $x$ will be the slope, $m$.

Detailed Solution:

1. Start with the general equation: $6x - 2y + 8 = 0$
2. Isolate the y-term: $-2y = -6x - 8$
3. Divide the entire equation by -2 to solve for y: $y = \frac{-6x}{-2} + \frac{-8}{-2}$
4. Simplify: $y = 3x + 4$
5. Identify the slope: The equation is now in $y = mx + c$ form. The slope $m$ is the coefficient of $x$.

Final Answer: The slope is 3.

Question 3: Converting Linear Equation Forms (from file image_0b62af.png)

The Question: If the general equation of a line is $3x + 5y - 30 = 0$, then choose the set of correct options. (Multiple Select Question)

• The intercept form of the line is $\frac{x}{10} + \frac{y}{6} = 1$
• The slope intercept form of the line is $y = -3x - 30$
• The intercept form of the line is $5x + 3y - 30 = 1$
• The slope intercept form of the line is $y = -\frac{3}{5}x + 6$

Core Concepts: Converting from General Form

• To Slope-Intercept Form ($y = mx + c$): Isolate the $y$ variable.
• To Intercept Form ($\frac{x}{a} + \frac{y}{b} = 1$): Move the constant term to the right side, then divide the entire equation by that constant to make the right side equal to 1.

Detailed Solution:

Let’s start with the given equation: $3x + 5y - 30 = 0$.

Part 1: Convert to Intercept Form

1. Move the constant term to the right side: $3x + 5y = 30$
2. Divide the entire equation by 30 to make the right side equal to 1: $\frac{3x}{30} + \frac{5y}{30} = \frac{30}{30}$
3. Simplify each fraction: $\frac{x}{10} + \frac{y}{6} = 1$
   • This matches the first option. The third option is incorrect.

Part 2: Convert to Slope-Intercept Form

1. Start with the general equation and isolate the y-term: $5y = -3x + 30$
2. Divide the entire equation by 5 to solve for y: $y = \frac{-3x}{5} + \frac{30}{5}$
3. Simplify: $y = -\frac{3}{5}x + 6$
   • This matches the fourth option. The second option is incorrect.

Final Answer: The correct options are “The intercept form of the line is $\frac{x}{10} + \frac{y}{6} = 1$” and “The slope intercept form of the line is $y = -\frac{3}{5}x + 6$”.

1) If the x-intercept and the y-intercept of a straight line are -6 and 7 respectively, then choose the correct equation of the line.

Concept:

• The intercept form of a line is:

where $ a $ is the x-intercept and $ b $ the y-intercept.

• Here, $ a = -6 $, $ b = 7 $.

Multiply both sides by 42 (LCM of denominators):

Bring all terms to one side:

(since -42 = 0 and +42 = 0 both point to the same structure but check choices!)

• This matches the option: $ 7x - 6y + 42 = 0 $.

Final Answer: $ \boxed{7x - 6y + 42 = 0} $

2) The slope of the line $ 6x - 2y + 8 = 0 $ is ______

Concept:

• The general equation of a line is $ Ax + By + C = 0 $.
• The slope $ m = -\frac{A}{B} $.

Apply it:

Final Answer: $ \boxed{3} $

3) If the general equation of a line is $ 3x + 5y - 30 = 0 $, then choose the set of correct options:

Concepts:

a) Intercept Form:

Rewrite as

So: $ \frac{x}{10} + \frac{y}{6} = 1 $ is correct.

b) Slope-Intercept Form:

So: $ \boxed{y = -\frac{3}{5}x + 6} $

c) Check Other Intercept Option:

This is not equivalent to our given line.

d) Check Other Slope-Intercept Option:

This does not match.

Final Answers:

• The intercept form of the line is $ \frac{x}{10} + \frac{y}{6} = 1 $
• The slope intercept form is $ y = -\frac{3}{5}x + 6 $

4) What is the slope of the line: $ 4x - 2y - 12 = 0 $?

Concept: General form $ Ax + By + C = 0 $, slope is $ m = -\frac{A}{B} $.

Final Answer: $ \boxed{2} $

5) What is the x-intercept?

Concept: Set $ y = 0 $, solve for $ x $:

Final Answer: $ \boxed{3} $

6) What is the y-intercept?

Concept: Set $ x = 0 $, solve for $ y $:

Final Answer: $ \boxed{-6} $