Find the Line Equation: Parallel to x+3y=4x+9 Through Point (6,14)

Q: Why can't I just use the coefficients 1 and 3 to find the slope?

Because the equation isn't in standard form yet! You have $ x + 3y = 4x + 9 $, not $ x + 3y = 9 $. Always simplify first to get the true relationship.

Q: How do I know if two lines are really parallel?

Two lines are parallel if they have exactly the same slope but different y-intercepts. Same slope means they'll never intersect!

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

Point-slope: $ y - y_1 = m(x - x_1) $ uses a known pointSlope-intercept: $ y = mx + b $ shows the y-intercept directly

Q: Can I check my answer without substituting the point?

Yes! Your line should have the same slope as the original line. If you get $ y = x + 8 $, the slope is 1, which matches the given line's slope.

Q: What if I get a different y-intercept than the original line?

That's exactly what should happen! Parallel lines have the same slope but different y-intercepts. If they had the same intercept, they'd be the same line!

Parallel Lines with Point-Slope Form

Straight line passes through the point $(6,14)$ and parallel to the line $x+3y=4x+9$

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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 Isolate Y

00:22 This is the equation of the parallel function

00:32 This is the slope of the parallel function

00:36 Parallel functions have equal slopes

00:43 Let's use the line equation

00:46 Let's substitute the point according to the given data

00:50 Let's substitute the slope and solve to find the intersection point (B)

01:03 Let's isolate the intersection point (B)

01:12 This is the intersection point with the Y-axis

01:16 Now let's substitute the intersection point and slope in the line equation

01:27 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

Straight line passes through the point $(6,14)$ and parallel to the line $x+3y=4x+9$

Step-by-step solution

To solve the problem of determining the equation of the line parallel to $x + 3y = 4x + 9$ and passing through point $(6, 14)$ , we will follow these steps:

Step 1: Rearrange the given line to find its slope.
Step 2: Use the slope and the given point to apply the point-slope form of a line.
Step 3: Convert the equation into the slope-intercept form for clarity.

Let's perform each step:

Step 1: First, the equation $x + 3y = 4x + 9$ simplifies to find the slope. Subtract $x$ from both sides to get $3y = 3x + 9$ . By dividing everything by 3, this gives $y = x + 3$ . Therefore, the slope, $m$ , is 1.

Step 2: Given that parallel lines share the same slope, the slope of the desired line is also 1. Applying the point-slope form: $y - 14 = 1(x - 6)$ .

Step 3: Simplifying this equation yields $y - 14 = x - 6$ . Further rearranging gives $y = x + 8$ .

Thus, the equation of the line parallel to the given line and passing through $(6, 14)$ is $y = x + 8$ .

Final Answer

$y=x+8$

Key Points to Remember

Essential concepts to master this topic

Rule: Parallel lines have identical slopes when in standard form
Technique: Rearrange $x + 3y = 4x + 9$ to get slope 1
Check: Substitute (6,14): 14 = 6 + 8 gives 14 = 14 ✓

Common Mistakes

Avoid these frequent errors

Using original equation coefficients as slope
Don't assume the slope is -1/3 from x + 3y = incorrect approach! This ignores the 4x term and gives wrong slope. Always rearrange to y = mx + b form first to find the true slope.

Practice Quiz

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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 the coefficients 1 and 3 to find the slope?

Because the equation isn't in standard form yet! You have $x + 3y = 4x + 9$ , not $x + 3y = 9$ . Always simplify first to get the true relationship.

How do I know if two lines are really parallel?

Two lines are parallel if they have exactly the same slope but different y-intercepts. Same slope means they'll never intersect!

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

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

Can I check my answer without substituting the point?

Yes! Your line should have the same slope as the original line. If you get $y = x + 8$ , the slope is 1, which matches the given line's slope.

What if I get a different y-intercept than the original line?

That's exactly what should happen! Parallel lines have the same slope but different y-intercepts. If they had the same intercept, they'd be the same line!

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