Q1: What is the slope-intercept form of a linear equation? (a) \(Ax + By = C\) (b) \(y = mx + b\) (c) \(y - y_1 = m(x - x_1)\) (d) \(x = my + b\)
Solution:
Ans: (b) Explanation: The slope-intercept form of a linear equation is \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept. Option (a) is standard form, option (c) is point-slope form, and option (d) is not a standard form of a linear equation.
Q2: A line has a slope of 3 and passes through the point (0, -2). What is the equation of the line in slope-intercept form? (a) \(y = 3x - 2\) (b) \(y = -2x + 3\) (c) \(y = 3x + 2\) (d) \(y = 2x - 3\)
Solution:
Ans: (a) Explanation: Using the slope-intercept form \(y = mx + b\), where \(m = 3\) and the y-intercept \(b = -2\) (since the line passes through (0, -2)), the equation is \(y = 3x - 2\).
Q3: Which form of a linear equation is represented by \(Ax + By = C\)? (a) Slope-intercept form (b) Point-slope form (c) Standard form (d) Vertex form
Solution:
Ans: (c) Explanation: The equation \(Ax + By = C\) is the standard form of a linear equation, where \(A\), \(B\), and \(C\) are constants and \(A\) and \(B\) are not both zero. Options (a) and (b) represent different forms, and option (d) is used for quadratic equations.
Q4: What is the slope of a line written in standard form as \(4x + 2y = 8\)? (a) 4 (b) 2 (c) -2 (d) -4
Solution:
Ans: (c) Explanation: To find the slope from standard form \(Ax + By = C\), use the formula \(m = -\frac{A}{B}\). Here, \(A = 4\) and \(B = 2\), so \(m = -\frac{4}{2} = -2\).
Q5: The point-slope form of a linear equation is \(y - y_1 = m(x - x_1)\). Which of the following represents this form for a line with slope 5 passing through the point (2, 3)? (a) \(y - 3 = 5(x - 2)\) (b) \(y - 2 = 5(x - 3)\) (c) \(y - 5 = 3(x - 2)\) (d) \(y - 3 = 2(x - 5)\)
Solution:
Ans: (a) Explanation: In point-slope form \(y - y_1 = m(x - x_1)\), substitute \(m = 5\), \(x_1 = 2\), and \(y_1 = 3\) to get \(y - 3 = 5(x - 2)\). The other options incorrectly swap or use wrong values for the slope and point coordinates.
Ans: (a) Explanation: Start with \(y - 4 = 2(x + 1)\). Distribute: \(y - 4 = 2x + 2\) Add 4 to both sides: \(y = 2x + 6\) The equation in slope-intercept form is \(y = 2x + 6\).
Q7: What is the y-intercept of the line represented by the equation \(3x - 6y = 12\)? (a) 2 (b) -2 (c) 4 (d) -4
Solution:
Ans: (b) Explanation: To find the y-intercept, set \(x = 0\): \(3(0) - 6y = 12\) \(-6y = 12\) \(y = -2\) The y-intercept is -2.
Q8: A line passes through the points (1, 2) and (3, 8). What is the slope of this line? (a) 2 (b) 3 (c) 4 (d) 6
Solution:
Ans: (b) Explanation: The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting the points (1, 2) and (3, 8): \(m = \frac{8 - 2}{3 - 1} = \frac{6}{2} = 3\)
Section B: Fill in the Blanks
Q9: In the slope-intercept form \(y = mx + b\), the variable \(m\) represents the __________.
Solution:
Ans: slope Explanation: In the equation \(y = mx + b\), \(m\) is the slope of the line, which measures the steepness and direction of the line.
Q10: In the slope-intercept form \(y = mx + b\), the variable \(b\) represents the __________.
Solution:
Ans: y-intercept Explanation: In the equation \(y = mx + b\), \(b\) is the y-intercept, which is the y-coordinate of the point where the line crosses the y-axis.
Q11: The standard form of a linear equation is \(Ax + By = C\), where \(A\), \(B\), and \(C\) are __________ and \(A\) should be a non-negative integer.
Solution:
Ans: integers (or constants) Explanation: In standard form, \(A\), \(B\), and \(C\) are typically integers (whole numbers), and by convention, \(A\) is non-negative.
Q12: The formula to calculate the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m =\) __________.
Solution:
Ans: \(\frac{y_2 - y_1}{x_2 - x_1}\) Explanation: The slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\) calculates the rate of change between two points on a line.
Q13: A horizontal line has a slope of __________.
Solution:
Ans: 0 (or zero) Explanation: A horizontal line has no vertical change, so its slope is zero. The equation of a horizontal line is \(y = k\), where \(k\) is a constant.
Q14: A vertical line has an __________ slope.
Solution:
Ans: undefined Explanation: A vertical line has an undefined slope because the change in \(x\) is zero, and division by zero is undefined. The equation of a vertical line is \(x = k\).
Section C: Word Problems
Q15: A taxi company charges a flat fee of $3 plus $2 per mile. Write an equation in slope-intercept form that represents the total cost \(C\) in dollars for a trip of \(m\) miles.
Solution:
Ans: The cost consists of a flat fee (y-intercept) and a variable cost per mile (slope). The flat fee is $3, and the cost per mile is $2. Using the slope-intercept form \(C = mx + b\) where \(C\) is the total cost and \(m\) is miles: \[C = 2m + 3\] Final Answer: \(C = 2m + 3\)
Q16: A line passes through the points (2, 5) and (6, 13). Find the equation of the line in slope-intercept form.
Solution:
Ans: First, calculate the slope using \(m = \frac{y_2 - y_1}{x_2 - x_1}\): \[m = \frac{13 - 5}{6 - 2} = \frac{8}{4} = 2\] Now use point-slope form with point (2, 5): \[y - 5 = 2(x - 2)\] Distribute: \(y - 5 = 2x - 4\) Add 5 to both sides: \(y = 2x + 1\) Final Answer: \(y = 2x + 1\)
Q17: The temperature in degrees Fahrenheit \(F\) is related to the temperature in degrees Celsius \(C\) by the equation \(F = \frac{9}{5}C + 32\). What is the slope of this equation and what does it represent?
Solution:
Ans: The equation is in slope-intercept form \(F = mC + b\). Comparing with \(F = \frac{9}{5}C + 32\), the slope is \(m = \frac{9}{5}\) or 1.8. The slope represents the rate of change of Fahrenheit with respect to Celsius. For every 1 degree increase in Celsius, Fahrenheit increases by \(\frac{9}{5}\) or 1.8 degrees. Final Answer: The slope is \(\frac{9}{5}\) (or 1.8), representing the rate of change of Fahrenheit per degree Celsius
Q18: Convert the equation \(5x + 2y = 10\) from standard form to slope-intercept form.
Solution:
Ans: Start with \(5x + 2y = 10\). Subtract \(5x\) from both sides: \(2y = -5x + 10\) Divide every term by 2: \(y = -\frac{5}{2}x + 5\) Final Answer: \(y = -\frac{5}{2}x + 5\)
Q19: A swimming pool contains 500 gallons of water. Water is being drained at a rate of 25 gallons per minute. Write an equation in slope-intercept form that models the amount of water \(W\) (in gallons) remaining in the pool after \(t\) minutes.
Solution:
Ans: The initial amount of water is 500 gallons (the y-intercept). The water is being drained at 25 gallons per minute, so the rate of change (slope) is -25. Using slope-intercept form \(W = mt + b\): \[W = -25t + 500\] Final Answer: \(W = -25t + 500\)
Q20: Write the equation of a line in point-slope form that has a slope of -3 and passes through the point (4, -1). Then convert it to slope-intercept form.
Solution:
Ans: Using point-slope form \(y - y_1 = m(x - x_1)\) with \(m = -3\), \(x_1 = 4\), and \(y_1 = -1\): \[y - (-1) = -3(x - 4)\] \[y + 1 = -3(x - 4)\] This is the point-slope form. Now convert to slope-intercept form: Distribute: \(y + 1 = -3x + 12\) Subtract 1 from both sides: \(y = -3x + 11\) Final Answer: Point-slope form: \(y + 1 = -3(x - 4)\); Slope-intercept form: \(y = -3x + 11\)
The document Worksheet (with Solutions): Forms Of Linear Equations is a part of the Grade 9 Course Mathematics: Algebra 1.
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