Find the Linear Function Equation: Slope 6 Through Point (1,1)

Q: Why can't I just use y = 6x since I know the slope is 6?

Because y = 6x means the line passes through (0,0), not (1,1)! You need the y-intercept to complete the equation. Use the given point to find where the line crosses the y-axis.

Q: How do I find the y-intercept from the slope and point?

Use the point-slope form first: $ y - 1 = 6(x - 1) $. When you solve for y, you get $ y = 6x - 5 $, so the y-intercept is -5.

Q: What if I get a different form like x - y = something?

That's fine too! You can rearrange $ y = 6x - 5 $ to get $ 6x - y = 5 $. Both represent the same line, just written differently.

Q: Why does the point-slope form work better than guessing?

Point-slope form guarantees your line passes through the given point! Guessing the y-intercept often leads to lines that miss the point entirely. Always use the formula for accuracy.

Point-Slope Form with Given Coordinates

A linear function with a slope of 6 passes through the point $(1,1)$ .

Which equation represents the 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

00:03 The given slope and point

00:13 Use the formula to represent a linear function

00:18 Substitute appropriate values according to the given data, and solve for B

00:32 Isolate the unknown B

00:39 This is the Y-axis intercept

00:48 Substitute the slope and intercept accordingly to find the function

01:01 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 6 passes through the point $(1,1)$ .

Which equation represents the function?

Step-by-step solution

To solve the problem of finding the equation of the linear function, we will use the point-slope form, which is:

y - y_1 = m(x - x_1)

Step-by-step:

Step 1: Identify given information: The slope $m = 6$ and the point $(x_1, y_1) = (1, 1)$ .
Step 2: Substitute the slope and point into the point-slope form:
$y - 1 = 6(x - 1)$
Step 3: Simplify the equation:
$y - 1 = 6x - 6$
Step 4: Solve for $y$ to express in slope-intercept form $y = mx + b$ :
$y = 6x - 6 + 1$
Step 5: Simplify the right-hand side:
$y = 6x - 5$

Thus, the equation of the linear function is $y = 6x - 5$ .

Final Answer

$y=6x-5$

Key Points to Remember

Essential concepts to master this topic

Point-Slope Formula: Use $y - y_1 = m(x - x_1)$ with given slope and point
Substitution: Replace m=6 and (1,1): $y - 1 = 6(x - 1)$
Verify: Check that point (1,1) satisfies final equation: $1 = 6(1) - 5 = 1$ ✓

Common Mistakes

Avoid these frequent errors

Using slope-intercept form y = mx + b without finding b first
Don't jump straight to y = 6x + b and guess the y-intercept = wrong equation like y = 6x + 1! You need the actual point to find b correctly. Always use point-slope form first, then convert to slope-intercept form.

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 = 6x since I know the slope is 6?

Because y = 6x means the line passes through (0,0), not (1,1)! You need the y-intercept to complete the equation. Use the given point to find where the line crosses the y-axis.

How do I find the y-intercept from the slope and point?

Use the point-slope form first: $y - 1 = 6(x - 1)$ . When you solve for y, you get $y = 6x - 5$ , so the y-intercept is -5.

Can I check my answer without graphing?

Yes! Substitute the given point (1,1) into your final equation. For $y = 6x - 5$ : does $1 = 6(1) - 5$ ? Yes, $1 = 1$ ✓

What if I get a different form like x - y = something?

That's fine too! You can rearrange $y = 6x - 5$ to get $6x - y = 5$ . Both represent the same line, just written differently.

Why does the point-slope form work better than guessing?

Point-slope form guarantees your line passes through the given point! Guessing the y-intercept often leads to lines that miss the point entirely. Always use the formula for accuracy.

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