Practice Problems

Problems:

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Problem 1

Explanation: When two lines are parallel, they have the exact same slope. This means they go in the same direction but never touch. If you know the slope of one line, then the slope of any line parallel to it is the same.

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Problem 2

Explanation: Perpendicular lines meet at a right angle ($90$ $degrees$). The slope of a perpendicular line is the negative reciprocal of the original slope. This means you flip the fraction and change the sign.

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Problem 3

Explanation: To see if two lines are parallel or perpendicular, compare their slopes. If the slopes are the same, the lines are parallel. If one slope is the negative reciprocal of the other, they're perpendicular. If neither, they're just different lines.

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Problem 4

Explanation: If you need to find the equation of a line that is perpendicular to another line, start by finding the slope of the perpendicular line. Then, use the point given and the slope to write the equation.

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Solutions:

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Problem 1

Solution:

When finding the slope of a line parallel to another, remember that parallel lines have the same slope. So, all we need to do is identify the slope of the given line.

Step 1: Identify the Slope

The given equation is in slope-intercept form, $y = mx + c$, where $m$ is the slope. In the equation $y = 3x + 2$, the slope $m = 3$.

Step 2: Conclusion

Since parallel lines have the same slope, the slope of any line parallel to this one will also be $m = 3$.

Answer: The slope of the parallel line is $3$.

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Problem 2

Solution:

To find the slope of a line that is perpendicular to another, remember the rule: perpendicular lines have slopes that multiply to give $-1$. This can also be remembered as "flip it and change the sign."

Step 1: Identify the Slope

The given equation is in slope-intercept form, $y = mx + c$, where $m$ is the slope. In the equation $y = -4x + 1$, the slope $m = -4$.

Step 2: Find the Perpendicular Slope

To find the slope of a line perpendicular to this one, we:

• Flip the slope: $-4$ becomes $\frac{1}{4}$.
• Change the sign: Since the slope was negative, the perpendicular slope will be positive. Answer: The slope of the perpendicular line is $\frac{1}{4}$.

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Problem 3

Solution:

To determine whether the lines are parallel, perpendicular, or neither, we compare their slopes.

Step 1: Identify the Slopes

Both equations are in slope-intercept form, $y = mx + c$.

• For the first line, $y = 2x - 5$, the slope $m = 2$.

• For the second line, $y = 2x + 3$, the slope $m = 2$. Step 2: Compare the Slopes

• Since both lines have the same slope $(m = 2)$, they are parallel. Answer: The lines are parallel because their slopes are equal.

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Problem 4

Solution:

To find the equation of a line that is perpendicular to another, we first find the perpendicular slope and then use the point-slope formula to write the equation of the line.

Step 1: Identify the Slope of the Given Line

The given equation is $y = \frac{1}{3}x + 4$. The slope of this line is $m = \frac{1}{3}$.

Step 2: Find the Perpendicular Slope

To find the slope of a perpendicular line:

• Flip the slope: $\frac{1}{3}$ becomes $3$.
• Change the sign: Since the slope was positive, the perpendicular slope will be negative. So, the perpendicular slope is $m = -3$.

Step 3: Use the Point-Slope Formula

Now that we know the perpendicular slope and have a point $(1, 2)$ on the line, we can use the point-slope formula to find the equation of the line:

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

Here, $m = -3$, $x_1 = 1$, and $y_1 = 2$.

Step 4: Substitute the Values

Substitute the values into the formula:

$y - 2 = -3(x - 1)$

Step 5: Simplify

• Distribute the $-3$ on the right side: $y - 2 = -3x + 3$

• Finally, add 2 to both sides to solve for $y$: $y = -3x + 5$

Answer: The equation of the line that passes through $(1, 2)$and is perpendicular to $y = \frac{1}{3}x + 4$ is $y = -3x + 5$.

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