Find the Line Equation Through Points (1/3,1) and (-1/3,2): Coordinate Geometry

Line Equations with Fractional Coordinates

Find the equation of the line passing through the two points (13,1),(−13,2) (\frac{1}{3},1),(-\frac{1}{3},2)

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Find the algebraic representation of the function
00:04 We'll use the formula to find the slope using 2 points
00:08 In each point, the left number represents X-axis and the right Y
00:16 We'll substitute the points according to the given data and find the slope
00:32 This is the line's slope
00:38 Now we'll use the line equation
00:43 We'll substitute the point according to the given data
00:47 We'll substitute the slope, and solve to find the intersection point (B)
01:03 We'll isolate the intersection point (B)
01:08 This is the intersection point with the Y-axis
01:14 Now we'll substitute the intersection point and slope in the line equation
01:23 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Find the equation of the line passing through the two points (13,1),(−13,2) (\frac{1}{3},1),(-\frac{1}{3},2)

2

Step-by-step solution

In the first step, we'll find the slope using the formula:

m=y2−y1x2−x1 m=\frac{y_2-y_1}{x_2-x_1}

We'll substitute according to the given points:

m=1−213−(−13) m=\frac{1-2}{\frac{1}{3}-(-\frac{1}{3})}

m=−123=−32 m=\frac{-1}{\frac{2}{3}}=-\frac{3}{2}

Now we'll choose point (13,1) (\frac{1}{3},1) and use the formula:

y=mx+b y=mx+b

1=−32×13+b 1=-\frac{3}{2}\times\frac{1}{3}+b

1=−12+b 1=-\frac{1}{2}+b

b=112 b=1\frac{1}{2}

We'll substitute the given data into the formula to find the equation of the line:

y=−32x+112 y=-\frac{3}{2}x+1\frac{1}{2}

3

Final Answer

y=−32x+112 y=-\frac{3}{2}x+1\frac{1}{2}

Key Points to Remember

Essential concepts to master this topic
  • Slope Formula: Use m=y2−y1x2−x1 m = \frac{y_2 - y_1}{x_2 - x_1} for any two points
  • Technique: Calculate 1−213−(−13)=−123=−32 \frac{1-2}{\frac{1}{3}-(-\frac{1}{3})} = \frac{-1}{\frac{2}{3}} = -\frac{3}{2}
  • Verification: Substitute both points into final equation to confirm they satisfy it ✓

Common Mistakes

Avoid these frequent errors
  • Incorrectly subtracting negative fractions in denominator
    Don't calculate 13−(−13) \frac{1}{3} - (-\frac{1}{3}) as 0 0 = undefined slope! This happens when you forget that subtracting a negative becomes addition. Always remember: 13−(−13)=13+13=23 \frac{1}{3} - (-\frac{1}{3}) = \frac{1}{3} + \frac{1}{3} = \frac{2}{3} .

Practice Quiz

Test your knowledge with interactive questions

Look at the function shown in the figure.

When is the function positive?

xy-4-7

FAQ

Everything you need to know about this question

Why do I get a negative slope when the line goes up from left to right?

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Actually, this line goes down from left to right! Point (−13,2) (-\frac{1}{3}, 2) is higher than point (13,1) (\frac{1}{3}, 1) , so as x increases, y decreases - that's a negative slope.

How do I subtract negative fractions correctly?

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Remember: subtracting a negative = adding a positive. So 13−(−13)=13+13=23 \frac{1}{3} - (-\frac{1}{3}) = \frac{1}{3} + \frac{1}{3} = \frac{2}{3} . Don't let the negative signs confuse you!

Can I use the other point to find the y-intercept?

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Absolutely! You should get the same answer. Try using (−13,2) (-\frac{1}{3}, 2) in y=mx+b y = mx + b : 2=−32(−13)+b 2 = -\frac{3}{2}(-\frac{1}{3}) + b gives b=112 b = 1\frac{1}{2} too!

What does the mixed number 112 1\frac{1}{2} mean in the equation?

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The mixed number 112 1\frac{1}{2} equals 32 \frac{3}{2} as an improper fraction. Both forms are correct - use whichever matches the answer choices!

How can I check if my final equation is right?

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Substitute both original points into your equation. For y=−32x+112 y = -\frac{3}{2}x + 1\frac{1}{2} : Point (13,1) (\frac{1}{3}, 1) gives 1=−12+32=1 1 = -\frac{1}{2} + \frac{3}{2} = 1 ✓

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