Find the Linear Function Equation: Slope 5 Through Point (2,4)

Q: Why can't I just use y = mx + b directly?

You could, but you'd need to find b first! The point-slope form is more direct when you have a point and slope. It automatically handles the algebra for you.

Q: How do I know which point to use as (x₁, y₁)?

Use the given point directly! In this problem, $ (x_1, y_1) = (2, 4) $. The point-slope formula works with any point on the line.

Q: Can I check my answer without substituting?

Yes! Check if your equation has the correct slope (5) and see if it makes sense. But substituting the point is the most reliable verification method.

Q: What's the difference between point-slope and slope-intercept form?

Point-slope: $ y - y_1 = m(x - x_1) $ (uses a specific point)Slope-intercept: $ y = mx + b $ (shows y-intercept directly)

Point-Slope Form with Slope-Intercept Conversion

A linear function with a slope of 5 passes through the point $(2,4)$ .

Choose the equation that represents this function.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find algebraic representation for the function

00:04 The given slope and point

00:09 Use the formula to represent a linear function

00:15 Substitute appropriate values according to the given data, and solve for B

00:34 Isolate the unknown B

00:48 Substitute accordingly the slope and intersection point to find the function

01:05 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 linear function with a slope of 5 passes through the point $(2,4)$ .

Choose the equation that represents this function.

Step-by-step solution

To solve for the equation of a line given a slope and a point:

Step 1: Use the point-slope form of the linear equation: $y - y_1 = m(x - x_1)$ .
Step 2: Substitute the given slope $m = 5$ and point $(2, 4)$ into the equation.
Step 3: Simplify and rearrange to convert into slope-intercept form $y = mx + b$ .

Substituting into the point-slope form, we have:

$y - 4 = 5(x - 2)$

Distribute the 5 across the terms in the parentheses:

$y - 4 = 5x - 10$

Add 4 to both sides to solve for $y$ :

$y = 5x - 10 + 4$

This simplifies to:

$y = 5x - 6$

Therefore, the equation of the line is $y = 5x - 6$ .

The correct choice among the options given is $y = 5x - 6$ .

Final Answer

$y=5x-6$

Key Points to Remember

Essential concepts to master this topic

Point-Slope Formula: Use $y - y_1 = m(x - x_1)$ when given slope and point
Substitution Technique: Replace m=5 and (2,4) to get $y - 4 = 5(x - 2)$
Verification Check: Substitute point (2,4) into final equation: $4 = 5(2) - 6 = 4$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting to distribute the slope correctly
Don't write $y - 4 = 5x - 2$ instead of $y - 4 = 5x - 10$ ! This happens when you forget to multiply 5 × 2 = 10. Always distribute the slope to both terms inside the parentheses completely.

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

Why can't I just use y = mx + b directly?

You could, but you'd need to find b first! The point-slope form is more direct when you have a point and slope. It automatically handles the algebra for you.

What if I get a different y-intercept than the answer choices?

Double-check your distribution step! Make sure you multiplied the slope by both terms in the parentheses: $5(x - 2) = 5x - 10$ , not $5x - 2$ .

How do I know which point to use as (x₁, y₁)?

Use the given point directly! In this problem, $(x_1, y_1) = (2, 4)$ . The point-slope formula works with any point on the line.

Can I check my answer without substituting?

Yes! Check if your equation has the correct slope (5) and see if it makes sense. But substituting the point is the most reliable verification method.

What's the difference between point-slope and slope-intercept form?

Point-slope: $y - y_1 = m(x - x_1)$ (uses a specific point)
Slope-intercept: $y = mx + b$ (shows y-intercept directly)

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