# Worksheet: Forms of Linear Equations ## Section A: Multiple Choice Questions
Q1: Which form of a linear equation is represented by \(y = mx + b\)? (a) Standard form (b) Slope-intercept form (c) Point-slope form (d) Vertex form
Solution:
Ans: (b) Explanation: The equation \(y = mx + b\) is the slope-intercept form of a linear equation, where \(m\) represents the slope and \(b\) represents the y-intercept. Standard form is \(Ax + By = C\), point-slope form is \(y - y_1 = m(x - x_1)\), and vertex form applies to quadratic equations.
Q2: What is the slope of the line represented by the equation \(3x + 4y = 12\)? (a) \(\frac{3}{4}\) (b) \(-\frac{3}{4}\) (c) \(\frac{4}{3}\) (d) 3
Solution:
Ans: (b) Explanation: To find the slope, convert to slope-intercept form: \(4y = -3x + 12\) \(y = -\frac{3}{4}x + 3\) The slope is the coefficient of \(x\), which is \(-\frac{3}{4}\). Option (a) has the wrong sign, option (c) inverts the fraction incorrectly, and option (d) ignores the \(y\)-coefficient.
Q3: The point-slope form of a linear equation is \(y - y_1 = m(x - x_1)\). Which of the following represents a line with slope 2 passing through the point (3, 5)? (a) \(y - 5 = 2(x - 3)\) (b) \(y - 3 = 2(x - 5)\) (c) \(y + 5 = 2(x + 3)\) (d) \(y - 2 = 5(x - 3)\)
Solution:
Ans: (a) Explanation: Using point-slope form \(y - y_1 = m(x - x_1)\) with \(m = 2\), \(x_1 = 3\), and \(y_1 = 5\), we get \(y - 5 = 2(x - 3)\). Option (b) swaps the coordinates, option (c) uses wrong signs, and option (d) swaps the slope and \(y_1\) value.
Q4: Which equation is in standard form? (a) \(y = 5x - 7\) (b) \(2x - 3y = 6\) (c) \(y + 4 = 3(x - 1)\) (d) \(x = 2y + 9\)
Solution:
Ans: (b) Explanation:Standard form is written as \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers and \(A\) is non-negative. Equation \(2x - 3y = 6\) fits this format. Option (a) is slope-intercept form, option (c) is point-slope form, and option (d) is not in standard form because the variable terms are not on the same side.
Q5: What is the y-intercept of the line \(y = -2x + 8\)? (a) -2 (b) 2 (c) 8 (d) -8
Solution:
Ans: (c) Explanation: In the slope-intercept form \(y = mx + b\), the y-intercept is represented by \(b\). In this equation, \(b = 8\), so the y-intercept is 8. Option (a) is the slope with wrong sign, option (b) is the absolute value of the slope, and option (d) has the wrong sign.
Ans: (a) Explanation: Distribute and simplify: \(y - 6 = 4x + 8\) \(y = 4x + 14\) Option (b) results from adding 6 to only the constant 8 without distributing, option (c) subtracts incorrectly, and option (d) makes an arithmetic error.
Q7: Which pair of lines are parallel? (a) \(y = 3x + 1\) and \(y = 3x - 5\) (b) \(y = 2x + 4\) and \(y = -2x + 4\) (c) \(y = \frac{1}{2}x + 3\) and \(y = 2x - 1\) (d) \(y = -x + 7\) and \(y = x + 7\)
Solution:
Ans: (a) Explanation:Parallel lines have the same slope but different y-intercepts. Both equations in option (a) have a slope of 3. Option (b) has opposite slopes, option (c) has reciprocal slopes, and option (d) has slopes that are opposites.
Q8: What is the x-intercept of the line \(5x - 2y = 10\)? (a) 2 (b) 5 (c) -5 (d) 10
Solution:
Ans: (a) Explanation: The x-intercept occurs when \(y = 0\): \(5x - 2(0) = 10\) \(5x = 10\) \(x = 2\) Option (b) is the coefficient of \(x\), option (c) is the negative of the coefficient, and option (d) is the constant term.
## Section B: Fill in the Blanks Q9: The equation \(Ax + By = C\) is called the __________ form of a linear equation.
Solution:
Ans: standard Explanation: The standard form of a linear equation is written as \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers.
Q10: In the slope-intercept form \(y = mx + b\), the variable \(m\) represents the __________.
Solution:
Ans: slope Explanation: In slope-intercept form, \(m\) is the coefficient of \(x\) and represents the slope of the line, which measures the steepness and direction of the line.
Q11: Two lines with the same slope but different y-intercepts are called __________ lines.
Solution:
Ans: parallel Explanation:Parallel lines never intersect and have identical slopes. They maintain a constant distance from each other throughout their length.
Q12: The point where a line crosses the y-axis is called the __________.
Solution:
Ans: y-intercept Explanation: The y-intercept is the value of \(y\) when \(x = 0\). In slope-intercept form \(y = mx + b\), this value is represented by \(b\).
Q13: The point-slope form of a linear equation is written as \(y - y_1 = m(x - __________)\).
Solution:
Ans: \(x_1\) Explanation: The point-slope form is \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a specific point on the line and \(m\) is the slope.
Q14: To convert from standard form to slope-intercept form, you must solve for the variable __________.
Solution:
Ans: \(y\) Explanation:Slope-intercept form requires the equation to be solved for \(y\), resulting in \(y = mx + b\). This makes it easy to identify the slope and y-intercept directly.
## Section C: Word Problems Q15: A taxi company charges a flat fee of $3.50 plus $0.75 per mile traveled. Write an equation in slope-intercept form that represents the total cost \(C\) for traveling \(m\) miles.
Solution:
Ans: \(C = 0.75m + 3.50\) Explanation: In slope-intercept form \(y = mx + b\), the slope represents the rate of change ($0.75 per mile) and the y-intercept represents the initial value (flat fee of $3.50). Final Answer: \(C = 0.75m + 3.50\)
Q16: A line passes through the points (2, 7) and (5, 16). Find the slope of this line.
Solution:
Ans: Using the slope formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\) \(m = \frac{16 - 7}{5 - 2} = \frac{9}{3} = 3\) Final Answer: The slope is 3
Q17: Write the equation of a line in point-slope form that passes through the point (-4, 3) and has a slope of -2.
Q18: Convert the equation \(6x + 3y = 18\) into slope-intercept form and identify the slope and y-intercept.
Solution:
Ans: Solve for \(y\): \(3y = -6x + 18\) \(y = -2x + 6\) The slope is -2 and the y-intercept is 6. Final Answer: \(y = -2x + 6\); slope = -2, y-intercept = 6
Q19: A pool contains 500 gallons of water and is being drained at a rate of 25 gallons per minute. Write an equation in slope-intercept form to represent the amount of water \(W\) in the pool after \(t\) minutes. How much water is in the pool after 12 minutes?
Solution:
Ans: The equation is \(W = -25t + 500\) After 12 minutes: \(W = -25(12) + 500\) \(W = -300 + 500\) \(W = 200\) gallons Final Answer: \(W = -25t + 500\); After 12 minutes, there are 200 gallons in the pool
Q20: Find the x-intercept and y-intercept of the line represented by the equation \(4x - 5y = 20\).
Solution:
Ans: For the x-intercept, set \(y = 0\): \(4x - 5(0) = 20\) \(4x = 20\) \(x = 5\) For the y-intercept, set \(x = 0\): \(4(0) - 5y = 20\) \(-5y = 20\) \(y = -4\) Final Answer: x-intercept = 5, y-intercept = -4
The document Worksheet (with Solutions): Forms of Linear Equations is a part of the Grade 9 Course Integrated Math 1.
past year papers, Worksheet (with Solutions): Forms of Linear Equations, practice quizzes, Summary, Worksheet (with Solutions): Forms of Linear Equations, Extra Questions, Worksheet (with Solutions): Forms of Linear Equations, Objective type Questions, MCQs, pdf , Important questions, mock tests for examination, Previous Year Questions with Solutions, study material, ppt, Semester Notes, video lectures, Free, shortcuts and tricks, Sample Paper, Exam, Viva Questions;
