Find The Slope Of The Line Whose Equation Is 5y = X - 3.5-3-3/51/5
Understanding how to determine the slope of a line from its equation is a fundamental skill in algebra and coordinate geometry. In this article, we will explore the process step-by-step to find the slope of the line given by the equation 5y = X - 3.5-3-3/51/5. We will analyze the equation, simplify it into slope-intercept form, and interpret the result. Whether you're a student preparing for exams or a math enthusiast brushing up on your skills, this comprehensive guide will clarify the process and enhance your understanding of linear equations.
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Deciphering the Given Equation
Before diving into calculations, it's crucial to interpret the provided equation accurately. The equation presented is:
5y = X - 3.5-3-3/51/5
At first glance, this appears somewhat confusing due to the mixture of numbers and fractions. Let's analyze it carefully.
Breaking Down the Equation
- The left side: 5y
- The right side: X - 3.5-3-3/51/5
The key is to understand what the right side represents. It appears to be an algebraic expression involving X and a series of constants and fractions.
Clarifying the Expression
The expression 3.5-3-3/51/5 can be interpreted as a sequence of subtractions and divisions. To avoid ambiguity, let's assume the expression is:
3.5 - 3 - (3/51) / 5
This is a reasonable interpretation because:
- The sequence 3.5 - 3 is straightforward.
- The next part 3/51 is a fraction.
- Then, dividing (3/51) by 5.
Alternatively, it could be a typo or misrepresentation, but for clarity, we'll proceed with this interpretation.
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Rewriting the Equation in a Clearer Format
To analyze and simplify the equation, it is best to write it explicitly:
5y = X - [3.5 - 3 - (3/51)/5]
Our goal is to express this equation in the slope-intercept form:
Y = mX + b
where m is the slope, and b is the y-intercept.
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Step-by-Step Simplification
Step 1: Simplify the Constants on the Right Side
Calculate the expression 3.5 - 3 - (3/51)/5
- First, compute 3.5 - 3
3.5 - 3 = 0.5
- Next, evaluate (3/51)/5
- Simplify 3/51
3/51 = 1/17 (since 3 ÷ 51 = 1 ÷ 17)
- Now, divide 1/17 by 5
(1/17) ÷ 5 = (1/17) × (1/5) = 1/85
- Now, substitute back into the expression:
0.5 - (1/85)
Step 2: Combine the Constants
Express 0.5 as a fraction to combine with 1/85:
- 0.5 = 1/2
- To subtract 1/85 from 1/2, find a common denominator:
- Denominator of 2 is 2
- Denominator of 85 is 85
- Common denominator: 170 (since 2 × 85 = 170)
- Convert both fractions:
- 1/2 = 85/170
- 1/85 = 2/170
- Subtract:
85/170 - 2/170 = (85 - 2)/170 = 83/170
Step 3: Rewrite the Equation
Now, the right side becomes:
X - 83/170
Returning to the original equation:
5y = X - 83/170
Step 4: Solve for y
Divide both sides by 5 to isolate y:
y = (X - 83/170) / 5
Express as:
y = (1/5)X - (83/170) / 5
Simplify the second term:
- Recall that dividing a fraction by a number is equivalent to multiplying numerator by 1 and denominator by that number:
(83/170) ÷ 5 = 83/170 × 1/5 = 83 / (170 × 5) = 83 / 850
Step 5: Final Slope-Intercept Form
Thus, the equation in slope-intercept form is:
y = (1/5)X - 83/850
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Identifying the Slope
In the slope-intercept form:
y = mX + b
- m is the slope of the line.
- b is the y-intercept.
From our simplified equation:
y = (1/5)X - 83/850
We see that:
m = 1/5
b = -83/850
Therefore, the slope of the line is 1/5.
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Understanding the Significance of the Slope
The slope of a line indicates how steep the line is and the direction it moves as x increases.
What Does a Slope of 1/5 Mean?
- For every 1 unit increase in X, Y increases by 1/5 units.
- The positive slope signifies that the line rises from left to right.
- The line is relatively gentle, with a modest upward incline.
Visual Representation
Imagine plotting the line on a coordinate plane:
- Starting at the y-intercept -83/850 (~ -0.0976), which is just below zero.
- For each step of 5 units along the X-axis, Y increases by 1 unit.
This understanding helps in graphing the line accurately.
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Additional Insights and Applications
How to Use the Slope in Problem Solving
Knowing the slope allows you to:
- Graph the line quickly by starting at the y-intercept and applying the slope.
- Find the equation of a line given a point and the slope.
- Determine whether two lines are parallel (same slope) or perpendicular (negative reciprocal slopes).
Practical Examples
- Graphing the Line
- Plot the y-intercept at -83/850.
- From this point, move 5 units right along the X-axis and 1 unit up.
- Continue to plot additional points for accuracy.
- Calculating Intersection Points
- Use the slope and intercept to find where the line crosses other lines or axes.
- Real-World Contexts
- The slope could represent rates such as speed, cost per item, or other proportional relationships.
Common Mistakes to Avoid
- Confusing the coefficient of Y or X in the original equation.
- Misinterpreting fractions or constants within the equation.
- Forgetting to divide through to achieve the slope-intercept form.
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Summary and Final Remarks
In this comprehensive analysis, we started with the complex equation:
5y = X - 3.5-3-3/51/5
By carefully interpreting, simplifying constants, and algebraically manipulating the equation, we expressed it in the standard slope-intercept form:
y = (1/5)X - 83/850
This revealed that the slope of the line is 1/5.
Understanding how to convert an equation into slope-intercept form is essential for analyzing and graphing lines efficiently. The positive slope indicates an upward trend as X increases, and the precise value enables accurate plotting and problem-solving.
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In conclusion, the slope of the line given by the equation 5y = X - 3.5-3-3/51/5 is 1/5. Mastering such conversions and interpretations enhances your algebraic skills and deepens your understanding of linear relationships in mathematics.
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Meta Keywords: slope of a line, linear equations, slope-intercept form, algebra, coordinate geometry, how to find slope, equation simplification, graphing lines, algebra tips
