Math Problem Statement
Solution
Let's go through the mathematical problems visible from the image.
Problem 1 (Top left corner):
Given the straight line is parallel to the straight line . Find the value of .
We know that two parallel lines have the same gradient (slope). So, we need to find the slope of both lines and set them equal to solve for .
- Rewrite both equations in the slope-intercept form :
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The first equation: So, the slope .
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The second equation: So, the slope .
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Since the lines are parallel, their slopes must be equal: Solving for :
So the value of is , which corresponds to option D.
Problem 11:
Express as a single fraction in its simplest form.
Steps:
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The second fraction can be simplified:
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Now, combine the fractions:
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Find the common denominator (which is ):
Thus, the simplified fraction is , which matches option C.
Problem 12:
Express as a single fraction in the simplest form.
Steps:
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Rewrite the fractions:
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Combine them under a common denominator :
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Simplify the numerator:
Thus, the simplified fraction is , which matches option A.
Problem 13:
Given , calculate the value of .
Steps:
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Cross-multiply:
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Expand both sides:
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Simplify and solve:
There is no solution for this equation, so it seems like there might be a mistake in the problem setup. However, if interpreted as the choices indicate, the closest value would be , which corresponds to option D.
Do you want me to proceed with more questions, or do you have any specific ones you'd like more detail on?
Here are 5 related questions:
- How do you find the equation of a line parallel to a given line?
- How do you simplify algebraic fractions with different denominators?
- What are the steps to solve linear equations by cross-multiplication?
- How do you handle equations with no solutions?
- What is the significance of the slope in parallel lines?
Tip: When simplifying fractions, always look for common factors between the numerator and the denominator to further reduce the expression.
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Math Problem Analysis
Mathematical Concepts
Algebra
Linear Equations
Parallel Lines
Formulas
Slope of a line: m = -\frac{A}{B} from the general form Ax + By = C
Theorems
Parallel lines have equal slopes
Suitable Grade Level
Grades 9-10
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