Determine the Linear Equation: Given Slope 1.5 and Point (3, 7.5)

Q: Why can't I just use y = mx + b since I know the slope?

You know the slope m = 1½, but you don't know the y-intercept b! The point-slope formula helps you find the equation using the point (3, 7½) instead of guessing b.

Q: How do I work with mixed numbers like 1½?

Convert mixed numbers to improper fractions first: 1½ = 3/2 and 7½ = 15/2. This makes calculations much easier and prevents errors.

Q: Which answer choice format should I expect?

Linear equations can be written in different forms: point-slope, slope-intercept, or factored. Compare your result to all choices by expanding or factoring as needed.

Point-Slope Form with Mixed Numbers

A line has a slope of $1\frac{1}{2}$ and passes through the point $(3,7\frac{1}{2})$ .

Which expression corresponds to the line?

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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

00:03 We'll use the straight line equation

00:09 We'll substitute the point according to the given data

00:15 We'll substitute the line's slope according to the given data

00:21 We'll continue solving to find the intersection point

00:35 We'll isolate intersection point B

00:41 This is the intersection point with the Y-axis

00:46 Now we'll substitute the intersection point and slope in the line equation

01:02 We'll factor out 1.5 from the parentheses

01:07 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

A line has a slope of $1\frac{1}{2}$ and passes through the point $(3,7\frac{1}{2})$ .

Which expression corresponds to the line?

Step-by-step solution

To solve the problem of finding the equation of the line:

Step 1: Identify the given information: slope $m = 1\frac{1}{2} = \frac{3}{2}$ , and point $(x_1, y_1) = (3, 7\frac{1}{2}) = (3, \frac{15}{2})$ .
Step 2: Use the point-slope formula $y - y_1 = m(x - x_1)$ .
Step 3: Substitute the given slope and point into the formula: $y - \frac{15}{2} = \frac{3}{2}(x - 3)$ .
Step 4: Distribute the slope on the right side: $y - \frac{15}{2} = \frac{3}{2}x - \frac{9}{2}$ .
Step 5: Add $\frac{15}{2}$ to both sides to solve for $y$ : $y = \frac{3}{2}x - \frac{9}{2} + \frac{15}{2}$ .
Step 6: Simplify the right side: $y = \frac{3}{2}x + 3$ .
Step 7: Compare this expression to the provided choices to find a match. The form is the same as choice $y=1\frac{1}{2}(x+2)$ , rewritten correctly as $y = \frac{3}{2}(x + 2)$ .

Therefore, the expression that corresponds to the line is $y = 1\frac{1}{2}(x+2)$ .

Final Answer

$y=1\frac{1}{2}(x+2)$

Key Points to Remember

Essential concepts to master this topic

Point-Slope Formula: Use y - y₁ = m(x - x₁) with given slope and point
Mixed Numbers: Convert 1½ to 3/2 and 7½ to 15/2 for calculations
Verify Answer: Substitute x = 3 into final equation: y = 3/2(3 + 2) = 7.5 ✓

Common Mistakes

Avoid these frequent errors

Using slope-intercept form y = mx + b directly
Don't jump to y = mx + b without finding b first = guessing the y-intercept! You'll likely pick the wrong b value since it's not given. Always use point-slope form y - y₁ = m(x - x₁) first, then convert if needed.

Practice Quiz

Test your knowledge with interactive questions

Look at the linear function represented in the diagram.

When is the function positive?

FAQ

Everything you need to know about this question

Why can't I just use y = mx + b since I know the slope?

You know the slope m = 1½, but you don't know the y-intercept b! The point-slope formula helps you find the equation using the point (3, 7½) instead of guessing b.

How do I work with mixed numbers like 1½?

Convert mixed numbers to improper fractions first: 1½ = 3/2 and 7½ = 15/2. This makes calculations much easier and prevents errors.

Which answer choice format should I expect?

Linear equations can be written in different forms: point-slope, slope-intercept, or factored. Compare your result to all choices by expanding or factoring as needed.

How do I check if my equation is correct?

Substitute the given point (3, 7½) into your equation. Both sides should equal the same value. For example: if y = 3/2(x + 2), then 7.5 = 3/2(3 + 2) = 3/2(5) = 7.5 ✓

What if my final equation looks different from the choices?

Algebraic expressions can look different but be equivalent! Try expanding, factoring, or converting between mixed numbers and improper fractions to match the format.

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