Determine the Equation of a Line Parallel to y=3x+4 Through (1/2,1)

Q: Why do parallel lines have the same slope?

Parallel lines never intersect, which means they rise and fall at exactly the same rate. The slope tells us this rate of change - if slopes were different, the lines would eventually cross!

Q: Can I just change the y-intercept and keep everything else?

No! You must calculate the new y-intercept using your given point. Just guessing or copying from the original line will give you the wrong equation.

Q: What if I get confused with the fractions?

Take it step by step! First substitute: $ y - 1 = 3(x - \frac{1}{2}) $. Then distribute: $ y - 1 = 3x - \frac{3}{2} $. Finally add 1: $ y = 3x - \frac{1}{2} $.

Q: How do I check if my answer is right?

Substitute your given point into your final equation! For $ (\frac{1}{2}, 1) $ in $ y = 3x - \frac{1}{2} $: 1 = 3(1/2) - 1/2 = 3/2 - 1/2 = 1 ✓

Q: Why can't the line be perpendicular instead?

The problem specifically asks for a parallel line! Perpendicular lines have slopes that are negative reciprocals (like 3 and -1/3), not the same slope.

Parallel Lines with Point-Slope Form

Given the line parallel to the line $y=3x+4$

and passes through the point $(\frac{1}{2},1)$ .

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

00:11 We'll use the line equation

00:15 We'll substitute the point according to the given data

00:19 We'll substitute the line's slope according to the given data

00:22 We'll continue solving to find the intersection point

00:31 We'll isolate the intersection point (B)

00:38 This is the intersection point with the Y-axis

00:43 Now we'll substitute the intersection point and slope in the line equation

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

Given the line parallel to the line $y=3x+4$

and passes through the point $(\frac{1}{2},1)$ .

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

Step-by-step solution

To solve this problem, we begin by noting that since the line is parallel to $y = 3x + 4$ , it must have the same slope, $m = 3$ .

We use the point-slope form of the equation of a line, which is:

y - y_1 = m(x - x_1)

Here, the slope $m = 3$ and the line passes through the point $\left(\frac{1}{2}, 1\right)$ . Therefore, we substitute these values into the point-slope formula:

y - 1 = 3\left(x - \frac{1}{2}\right)

Next, we simplify this equation:

Distribute the slope $3$ on the right side:

y - 1 = 3x - \frac{3}{2}

Add 1 to both sides to solve for $y$ :

y = 3x - \frac{3}{2} + 1

Simplify $-\frac{3}{2} + 1$ :

y = 3x - \frac{1}{2}

Thus, the equation of the line parallel to $y = 3x + 4$ and passing through the point $\left(\frac{1}{2}, 1\right)$ is:

y = 3x - \frac{1}{2}

The corresponding choice is:

$y=3x-\frac{1}{2}$

Final Answer

$y=3x-\frac{1}{2}$

Key Points to Remember

Essential concepts to master this topic

Parallel Lines: Same slope means equations have identical coefficients for x
Point-Slope Method: Use $y - 1 = 3(x - \frac{1}{2})$ with given point
Verification: Check that $y = 3(\frac{1}{2}) - \frac{1}{2} = 1$ ✓

Common Mistakes

Avoid these frequent errors

Using the y-intercept from the original line
Don't copy the +4 from y = 3x + 4 into your new equation = wrong line! This creates a line that doesn't pass through your given point. Always use the point-slope form with your specific point to find the correct y-intercept.

Practice Quiz

Test your knowledge with interactive questions

Look at the linear function represented in the diagram.

When is the function positive?

FAQ

Everything you need to know about this question

Why do parallel lines have the same slope?

Parallel lines never intersect, which means they rise and fall at exactly the same rate. The slope tells us this rate of change - if slopes were different, the lines would eventually cross!

Can I just change the y-intercept and keep everything else?

No! You must calculate the new y-intercept using your given point. Just guessing or copying from the original line will give you the wrong equation.

What if I get confused with the fractions?

Take it step by step! First substitute: $y - 1 = 3(x - \frac{1}{2})$ . Then distribute: $y - 1 = 3x - \frac{3}{2}$ . Finally add 1: $y = 3x - \frac{1}{2}$ .

How do I check if my answer is right?

Substitute your given point into your final equation! For $(\frac{1}{2}, 1)$ in $y = 3x - \frac{1}{2}$ : 1 = 3(1/2) - 1/2 = 3/2 - 1/2 = 1 ✓

Why can't the line be perpendicular instead?

The problem specifically asks for a parallel line! Perpendicular lines have slopes that are negative reciprocals (like 3 and -1/3), not the same slope.

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