Find the Equation of Line Parallel to y=2x+5 Through Point (4,9)

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

Parallel lines have identical slopes! From $ y = 2x + 5 $, the slope is 2, so any parallel line must also have slope 2.

Q: What's the difference between parallel and perpendicular lines?

Parallel lines have the same slope (like 2 and 2). Perpendicular lines have slopes that are negative reciprocals (like 2 and $ -\frac{1}{2} $).

Q: Can I use slope-intercept form directly?

You could, but point-slope form is easier when you have a specific point! Start with $ y - y_1 = m(x - x_1) $, then convert to slope-intercept form.

Q: Why don't parallel lines have the same y-intercept?

Parallel lines have the same slope but different y-intercepts. If they had the same y-intercept too, they'd be the exact same line, not just parallel!

Q: How can I check my answer is correct?

Substitute the given point into your equation! For $ y = 2x + 1 $: $ 9 = 2(4) + 1 = 9 $ ✓ It works!

Parallel Lines with Point-Slope Form

Given the line parallel to the line $y=2x+5$

and passes through the point $(4,9)$ .

Which of the algebraic representations is the corresponding one for the given 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 This is the slope of the line

00:08 Parallel lines have identical slopes

00:12 Let's use the line equation

00:16 Let's substitute the point according to the given data

00:25 Let's substitute the line slope according to the given data

00:31 Let's continue solving to find the intersection point

00:37 Let's isolate the intersection point (B)

00:40 This is the intersection point with the Y-axis

00:44 Now let's substitute the intersection point and slope in the line equation

00:57 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

Given the line parallel to the line $y=2x+5$

and passes through the point $(4,9)$ .

Which of the algebraic representations is the corresponding one for the given line?

Step-by-step solution

To solve this problem, let's proceed through these steps:

Step 1: Identify the slope of the original line. From $y = 2x + 5$ , the slope $m$ is $2$ .
Step 2: Since parallel lines have the same slope, the line we're looking for will also have a slope of $2$ .
Step 3: Use the point-slope formula with the given point $(4, 9)$ :

We begin with the point-slope formula:

$y - y_1 = m(x - x_1)$

Substitute $m = 2$ , $x_1 = 4$ , and $y_1 = 9$ into the equation:

$y - 9 = 2(x - 4)$

Simplify the equation:

$y - 9 = 2x - 8$

Solving for $y$ , we obtain:

$y = 2x - 8 + 9$

$y = 2x + 1$

Therefore, the algebraic representation of the line parallel to $y = 2x + 5$ that passes through $(4, 9)$ is:

$y = 2x + 1$

Final Answer

$y=2x+1$

Key Points to Remember

Essential concepts to master this topic

Parallel Lines: Always have identical slopes from original equation
Point-Slope: Use $y - 9 = 2(x - 4)$ with given point
Verify: Check that $(4,9)$ satisfies $y = 2x + 1$ ✓

Common Mistakes

Avoid these frequent errors

Using perpendicular slope instead of parallel slope
Don't use the negative reciprocal slope like $-\frac{1}{2}$ = wrong line type! Perpendicular lines use negative reciprocals, but parallel lines need the exact same slope. Always keep the original slope of 2 unchanged.

Practice Quiz

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Which statement best describes the graph below?

FAQ

Everything you need to know about this question

How do I know if two lines are parallel?

Parallel lines have identical slopes! From $y = 2x + 5$ , the slope is 2, so any parallel line must also have slope 2.

What's the difference between parallel and perpendicular lines?

Parallel lines have the same slope (like 2 and 2). Perpendicular lines have slopes that are negative reciprocals (like 2 and $-\frac{1}{2}$ ).

Can I use slope-intercept form directly?

You could, but point-slope form is easier when you have a specific point! Start with $y - y_1 = m(x - x_1)$ , then convert to slope-intercept form.

Why don't parallel lines have the same y-intercept?

Parallel lines have the same slope but different y-intercepts. If they had the same y-intercept too, they'd be the exact same line, not just parallel!

How can I check my answer is correct?

Substitute the given point into your equation! For $y = 2x + 1$ : $9 = 2(4) + 1 = 9$ ✓ It works!

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