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Quizzes > High School Quizzes > Mathematics

Point Slope Word Problems Practice Quiz

Master Equations Through Practical, Engaging Examples

Difficulty: Moderate
Grade: Grade 8
Study OutcomesCheat Sheet
Paper art promoting Point Slope Mastery, an interactive algebra quiz for high school students.

What is the point-slope form of a linear equation?
y - y₝ = m(x - x₝)
y = mx + b
x - x₝ = m(y - y₝)
y = m/x + c
The point-slope form of a linear equation is given by y - y₝ = m(x - x₝), where (x₝, y₝) is a point on the line and m is the slope. This format is very useful when you know a specific point and the slope of the line.
Find the equation in point-slope form that passes through (3, 4) with a slope of 2.
y - 4 = 2(x - 3)
y - 3 = 2(x - 4)
y + 4 = 2(x + 3)
y = 2x + 4
Using the point-slope form y - y₝ = m(x - x₝) and substituting the point (3, 4) with m = 2 gives y - 4 = 2(x - 3). This is the correct transformation according to the formula.
Which of the following is an equivalent form of the equation y - 5 = -3(x + 2)?
y = -3x - 1
y = -3x - 11
y = 3x + 1
y = -3x + 5
Expanding the equation gives y - 5 = -3x - 6, and by adding 5 to both sides, we obtain y = -3x - 1. This confirms that the equivalent slope-intercept form is option A.
Identify the slope and point in the equation: y - 7 = 4(x - 2).
Slope is 4 and point is (2, 7)
Slope is 7 and point is (4, 2)
Slope is 4 and point is (7, 2)
Slope is 2 and point is (7, 4)
The point-slope form y - y₝ = m(x - x₝) directly shows that m = 4 and the reference point (x₝, y₝) is (2, 7). This makes option A the correct interpretation.
What does the slope represent in the point-slope form of a line?
The rate of change of y with respect to x
The y-intercept of the line
The x-intercept of the line
The distance between two points
In any linear equation, the slope indicates the rate at which y changes with respect to x. It tells you how much y increases or decreases as x increases, making option A the correct choice.
Given the point-slope equation y + 2 = 3(x - 1), what is the slope-intercept form of the line?
y = 3x - 5
y = 3x + 5
y = 3x - 2
y = 3x + 2
First, distribute the 3 on the right side to obtain y + 2 = 3x - 3. Subtracting 2 from both sides gives y = 3x - 5, which is the required slope-intercept form.
Write the equation in point-slope form for a line that passes through (-4, 8) with a slope of -2.
y - 8 = -2(x + 4)
y + 8 = -2(x - 4)
y - 8 = 2(x + 4)
y + 8 = 2(x - 4)
Using the point-slope formula y - y₝ = m(x - x₝) with (x₝, y₝) = (-4, 8) and m = -2, we get y - 8 = -2(x - (-4)), which simplifies to y - 8 = -2(x + 4). This corresponds with option A.
Convert the point-slope form y - 3 = 0.5(x + 6) into slope-intercept form.
y = 0.5x + 6
y = 0.5x - 6
y = 0.5x + 3
y = 2x + 3
Distribute 0.5 on the right to get y - 3 = 0.5x + 3. Then add 3 to both sides by isolating y to obtain y = 0.5x + 6. Option A is correct.
Determine the missing value in the point-slope equation: y - ? = 4(x - 5) if the line passes through (5, 9).
y - 9 = 4(x - 5)
y - 5 = 4(x - 9)
y - 4 = 4(x - 5)
y - 5 = 4(x - 4)
In the point-slope form y - y₝ = m(x - x₝), the term (x - 5) indicates that x₝ = 5. Since the point is (5, 9), y₝ must be 9. Therefore, the correct equation is y - 9 = 4(x - 5).
If the equation in point-slope form is y - 2 = -3(x - 1), what is the y-intercept of the line?
5
2
-3
1
To find the y-intercept, convert the equation to slope-intercept form. Expanding gives y - 2 = -3x + 3 and then y = -3x + 5, so the y-intercept (the constant term) is 5.
For the line represented in point-slope form by y - 10 = (1/2)(x - 2), what is the slope?
1/2
2
10
-1/2
In the point-slope format y - y₝ = m(x - x₝), the coefficient m is directly the slope. Here m is 1/2, making option A the correct answer.
Which step is necessary to convert a given point-slope equation to slope-intercept form?
Distribute the slope and then add or subtract to isolate y
Multiply both sides by x
Divide both sides by the slope
Set x equal to zero
The conversion process involves distributing the slope through the parentheses and then isolating y by adding or subtracting the necessary constant. This reveals the slope-intercept form of the equation.
Solve the word problem: A car rental cost is modeled by the equation y - 50 = 20(x - 1), where x is the number of days and y is the total cost in dollars. What is the total cost for 4 days?
$110
$80
$90
$100
Substitute x = 4 into the equation: y - 50 = 20(4 - 1) becomes y - 50 = 60, so y = 110. The total cost for 4 days is $110, which is option A.
A line is defined by the point-slope equation y - 6 = -2(x - 3). What does the number 6 represent in this equation?
The y-coordinate of the point used
The slope of the line
The x-coordinate of the point used
The y-intercept of the line
In the point-slope form y - y₝ = m(x - x₝), the subtracted value from y is the y-coordinate of the specific point on the line. Thus, 6 represents the y-coordinate, making option A correct.
A snowboard rental is represented by y - 30 = 15(x - 2). If the rental cost increases linearly after 2 hours, what is the base fee?
$30
$15
$2
$45
In the point-slope form, the number subtracted from y indicates the y-coordinate of the reference point. Here it is 30, which suggests that the base fee before extra charges is $30, making option A correct.
Find the point-slope form of a line that passes through the intersection of the lines y = 2x + 3 and y = -x + 9, and has a slope of 1.
y - 7 = 1(x - 2)
y - 2 = 1(x - 7)
y - 7 = 1(x + 2)
y - 7 = -1(x - 2)
First, find the intersection of y = 2x + 3 and y = -x + 9 by solving 2x + 3 = -x + 9, which gives x = 2 and y = 7. Using the point-slope form with the point (2, 7) and slope 1 results in y - 7 = 1(x - 2).
Given the point-slope form y - k = m(x - h), if a line has a y-intercept of -4 and passes through (0, -4), which of the following represents its point-slope form with a slope of 3?
y + 4 = 3x
y - 4 = 3(x - 0)
y + 4 = 3(x + 0)
y - 4 = 3x
Since the line passes through (0, -4), the point to use is (0, -4). Plugging into the point-slope form, we have y - (-4) = 3(x - 0), which simplifies to y + 4 = 3x. This confirms option A as correct.
The point-slope form y - 9 = a(x - 4) represents a line that goes through (4, 9). If the line is parallel to the line given by y - 5 = 2(x - 1), what is the value of a?
2
-2
9
4
Parallel lines share the same slope. The slope of the line y - 5 = 2(x - 1) is 2, so the slope a in the equation y - 9 = a(x - 4) must also be 2. Therefore, option A is correct.
A line in point-slope form is given by y - b = m(x - 3) and passes through the point (3, b). When rewritten in slope-intercept form, the line passes through (0, c). How do you express c in terms of m and b?
c = b - 3m
c = 3m + b
c = m - 3b
c = 3b - m
Expanding the point-slope form gives y = m(x - 3) + b, which simplifies to y = mx - 3m + b. The y-intercept is the constant term, so c = b - 3m. Option A is the correct relationship.
Solve the word problem: A cell phone plan charges a fixed fee plus a per-minute rate. The cost for 10 minutes is $25 and the cost for 25 minutes is $40. Write the equation of cost in point-slope form using the 10-minute point and identify the per-minute rate.
Equation: C - 25 = 1(x - 10); per-minute rate: $1
Equation: C - 15 = 1(x - 10); per-minute rate: $1
Equation: C - 25 = (x - 10)/1; per-minute rate: $10
Equation: C - 25 = 1(x - 10); per-minute rate: $10
Let the fixed fee be F and the per-minute rate be r. Using the points (10, 25) and (25, 40), subtracting the equations gives 15r = 15, so r = 1 and F = 25 - 10 = 15. Using the 10-minute point in point-slope form yields C - 25 = 1(x - 10), and the rate is $1 per minute, which matches option A.
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Study Outcomes

  1. Understand the structure and key components of point-slope equations.
  2. Apply the point-slope formula to solve algebraic problems.
  3. Analyze word problems to extract relevant information such as slopes and coordinates.
  4. Evaluate detailed feedback to refine problem solving techniques for point-slope equations.

Point Slope Word Problems Cheat Sheet

  1. Understand the point-slope form equation - The point-slope form, y−y₝=m(x−x₝), links a known point and slope to build your line equation in a flash. It's perfect for graphing when you already have a point on the line and the slope handy. Math Is Fun: Point‑Slope Form
    2. Math Is Fun: Point‑Slope Form
  2. Calculate the slope from two points - The slope m tells you how steep your line climbs or falls, and it's just Δy/Δx. Grab two points (x₝,y₝) and (x₂,y₂), subtract y's and x's, and voilà - you've got your slope! GeeksforGeeks: Point‑Slope Formula
    3. GeeksforGeeks: Point‑Slope Formula
  3. Convert to slope-intercept form - Transforming to y=mx+b reveals the y-intercept b, which tells you where the line crosses the vertical axis. By isolating y in your point-slope equation, you'll master both forms in no time. Math Is Fun: Slope‑Intercept Form
    4. Math Is Fun: Slope‑Intercept Form
  4. See why point-slope is so useful - When you know one point and the slope, this form wastes no time on extra algebra. It's your secret weapon for quick graph sketches and problem-solving spurts. Math.net: Point‑Slope Form
    5. Math.net: Point‑Slope Form
  5. Handle vertical lines separately - Vertical lines can't use point-slope because their slope is undefined. Instead, remember that any vertical line is simply x = constant! Math Is Fun: Vertical Lines
    6. Math Is Fun: Vertical Lines
  6. Practice writing your own equations - Pick random points and slopes to challenge yourself - you'll quickly gain confidence. The more you try, the more natural point-slope form will feel! MathPlanet: Point‑Slope Practice
    7. MathPlanet: Point‑Slope Practice
  7. Discover its origin from the slope formula - Point-slope form is just a clever rearrangement of m = (y₂−y₝)/(x₂−x₝). Understanding the derivation helps cement why it works every time. Math.net: Deriving Point‑Slope
    8. Math.net: Deriving Point‑Slope
  8. Identify slope and point in a snap - Read m and (x₝,y₝) straight from the equation to graph without fuss. It's like decoding a secret message that points you right to the line! ChiliMath: Point‑Slope Form
    9. ChiliMath: Point‑Slope Form
  9. Model real-world scenarios - Use point-slope to translate word problems - like speed over time - into crisp equations. It's a powerful way to connect algebra with everyday life. GeeksforGeeks: Real‑World Applications
    10. GeeksforGeeks: Real‑World Applications
  10. Transform into standard form - Rearrange your point-slope equation into Ax+By=C to match any problem's requirements. This flexibility makes you the algebra superstar of any classroom! MathPlanet: Standard Form
    11. MathPlanet: Standard Form
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