Linear Function Equation: Finding the Line with Slope 1 Through (6,13)

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

You can use $ y = mx + b $, but you need to find b first! Substitute the given point $ (6,13) $ and slope $ m = 1 $: $ 13 = 1(6) + b $, so $ b = 7 $.

Q: What's the difference between the y-coordinate and y-intercept?

The y-coordinate is the second number in a point like $ (6,13) $ - here it's 13. The y-intercept is where the line crosses the y-axis (when x = 0). These are usually different values!

Q: How do I remember the point-slope formula?

Think of it as: "The change in y equals the slope times the change in x." So $ y - y_1 = m(x - x_1) $ means the vertical distance equals slope times horizontal distance.

Q: Can I check my answer a different way?

Yes! Pick any point on your line equation. For $ y = x + 7 $, when $ x = 0 $, $ y = 7 $. Check: slope from $ (0,7) $ to $ (6,13) $ is $ \frac{13-7}{6-0} = 1 $ ✓

Q: What if the slope was negative or a fraction?

The process is exactly the same! Just be extra careful with signs and fractions. For example, with slope $ -\frac{1}{2} $: $ y - 13 = -\frac{1}{2}(x - 6) $.

Point-Slope Form with Given Coordinates

A linear function has a slope of 1 and passes through the point $(6,13)$ .

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

00:03 The given slope and point

00:10 We'll use the formula for representing a linear function

00:14 We'll substitute appropriate values according to the given data, and solve to find B

00:26 We'll isolate the unknown B

00:34 This is the Y-axis intersection point (the unknown B)

00:38 We'll appropriately substitute the slope and intersection point to find the function

00:55 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 has a slope of 1 and passes through the point $(6,13)$ .

Choose the equation that represents this function.

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Use the point-slope form with the given point and slope.
Step 2: Simplify the equation to slope-intercept form.
Step 3: Identify the correct equation from the options.

Now, let's work through each step:

Step 1: Apply the point-slope form formula:
Given the point $(6, 13)$ and the slope $m = 1$ , the point-slope form is:
$y - 13 = 1(x - 6)$

Step 2: Simplify to get the equation in slope-intercept form:
$y - 13 = 1 \cdot (x - 6) \\ y - 13 = x - 6 \\ y = x - 6 + 13 \\ y = x + 7$

Step 3: Compare to find the correct answer:
From the simplified equation $y = x + 7$ , the correct choice is:

$y=x+7$

Therefore, the equation representing the function is $y = x + 7$ .

Final Answer

$y=x+7$

Key Points to Remember

Essential concepts to master this topic

Point-Slope Formula: Use $y - y_1 = m(x - x_1)$ with given point and slope
Substitution: Replace with $(6,13)$ and $m = 1$ to get $y - 13 = 1(x - 6)$
Verification: Check by substituting $(6,13)$ into $y = x + 7$ : $13 = 6 + 7$ ✓

Common Mistakes

Avoid these frequent errors

Using slope-intercept form directly without finding y-intercept
Don't assume $y = x + 13$ just because the point is $(6,13)$ = wrong y-intercept! The 13 is the y-coordinate of the given point, not the y-intercept. Always use point-slope form first, then simplify to find the correct y-intercept.

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 can use $y = mx + b$ , but you need to find b first! Substitute the given point $(6,13)$ and slope $m = 1$ : $13 = 1(6) + b$ , so $b = 7$ .

What's the difference between the y-coordinate and y-intercept?

The y-coordinate is the second number in a point like $(6,13)$ - here it's 13. The y-intercept is where the line crosses the y-axis (when x = 0). These are usually different values!

How do I remember the point-slope formula?

Think of it as: "The change in y equals the slope times the change in x." So $y - y_1 = m(x - x_1)$ means the vertical distance equals slope times horizontal distance.

Can I check my answer a different way?

Yes! Pick any point on your line equation. For $y = x + 7$ , when $x = 0$ , $y = 7$ . Check: slope from $(0,7)$ to $(6,13)$ is $\frac{13-7}{6-0} = 1$ ✓

What if the slope was negative or a fraction?

The process is exactly the same! Just be extra careful with signs and fractions. For example, with slope $-\frac{1}{2}$ : $y - 13 = -\frac{1}{2}(x - 6)$ .

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