Find the Line Equation Through Points (5,0) and (1/2,4.5): Coordinate Method

Q: How do I handle the mixed number 4½ in calculations?

Convert 4½ to 4.5 (as a decimal) or 9/2 (as an improper fraction). Both work perfectly for slope calculations - just stay consistent with your choice!

Q: Why is the answer y + x = 5 instead of y = -x + 5?

Both forms are correct! y = -x + 5 is slope-intercept form, while y + x = 5 is standard form. You can rearrange: y = -x + 5 → y + x = 5 by adding x to both sides.

Q: What if I get the slope as a fraction instead of -1?

Check your arithmetic! With points (5,0) and (0.5, 4.5): $ \frac{0-4.5}{5-0.5} = \frac{-4.5}{4.5} = -1 $. The numbers divide evenly to give exactly -1.

Q: How do I know which point to use for finding b?

You can use either point! Both (5,0) and (0.5, 4.5) will give you b = 5 when you substitute into y = mx + b with m = -1.

Q: Can I check my answer using both points?

Absolutely! For y + x = 5: Point (5,0) gives 0 + 5 = 5 ✓, and point (0.5, 4.5) gives 4.5 + 0.5 = 5 ✓. Both should work!

Slope-Intercept Form with Mixed Number Coordinates

Find the equation of the line passing through the two points $(5,0),(\frac{1}{2},4\frac{1}{2})$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:08 Let's find the algebraic representation of the function, shall we?

00:12 First, we're going to use a formula to determine the slope using two points. Ready?

00:18 At each point, remember, the left number is the X-axis and the right is the Y-axis. Easy, right?

00:26 Now, let's substitute these points into the formula with the data given, and find the slope. Let's do it!

00:42 Great! This number is the line's slope.

00:49 Next, we're going to use the line equation.

00:53 Substitute the point in the equation, following the data provided. You've got this!

00:59 Now, substitute the slope and solve for the intersection point, which we'll call 'B'. Keep going!

01:06 Let's isolate the intersection point, 'B'. Almost there!

01:12 And here is the intersection point with the Y-axis. Awesome job!

01:17 Now, plug this intersection point and the slope back into the line equation. You're doing amazing!

01:32 Let's tidy up the equation. Final steps!

01:39 And there you have it! That's the solution to our question. Well done!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Find the equation of the line passing through the two points $(5,0),(\frac{1}{2},4\frac{1}{2})$

Step-by-step solution

First, we will use the formula to find the slope of the straight line:

We replace the data and solve:

$\frac{(0-4.5)}{(5-0.5)}=\frac{-4.5}{4.5}=-1$

Now, we know that the slope is $-1$

We replace one of the points in the formula of the line equation:

$y=mx+b$

$(5,0)$

$0=-1\times5+b$

$0=-5+b$

$b=5$

Now we have the data to complete the equation:

$y=-1\times x+5$

$y=-x+5$

Final Answer

$y+x=5$

Key Points to Remember

Essential concepts to master this topic

Slope Formula: Use m = (y₂ - y₁)/(x₂ - x₁) with careful fraction conversion
Technique: Convert 4½ to 4.5, then calculate: (0-4.5)/(5-0.5) = -4.5/4.5 = -1
Check: Substitute both points into final equation y = -x + 5 ✓

Common Mistakes

Avoid these frequent errors

Converting mixed numbers incorrectly
Don't write 4½ as 4.2 or forget to convert it = wrong slope calculation! Mixed numbers must be converted properly: 4½ = 4.5 = 9/2. Always convert mixed numbers to decimals or improper fractions before calculating slope.

Practice Quiz

Test your knowledge with interactive questions

What is the solution to the following inequality?

$$ 10x-4\leq-3x-8 $$

FAQ

Everything you need to know about this question

How do I handle the mixed number 4½ in calculations?

Convert 4½ to 4.5 (as a decimal) or 9/2 (as an improper fraction). Both work perfectly for slope calculations - just stay consistent with your choice!

Why is the answer y + x = 5 instead of y = -x + 5?

Both forms are correct! y = -x + 5 is slope-intercept form, while y + x = 5 is standard form. You can rearrange: y = -x + 5 → y + x = 5 by adding x to both sides.

What if I get the slope as a fraction instead of -1?

Check your arithmetic! With points (5,0) and (0.5, 4.5): $\frac{0-4.5}{5-0.5} = \frac{-4.5}{4.5} = -1$ . The numbers divide evenly to give exactly -1.

How do I know which point to use for finding b?

You can use either point! Both (5,0) and (0.5, 4.5) will give you b = 5 when you substitute into y = mx + b with m = -1.

Can I check my answer using both points?

Absolutely! For y + x = 5: Point (5,0) gives 0 + 5 = 5 ✓, and point (0.5, 4.5) gives 4.5 + 0.5 = 5 ✓. Both should work!

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