Q1: Which of the following ordered pairs is a solution to the system of equations: \(y = 2x + 3\) and \(y = -x + 9\)? (a) (1, 5) (b) (2, 7) (c) (3, 6) (d) (4, 11)
Solution:
Ans: (b) Explanation: To find the solution, we substitute each ordered pair into both equations. For (2, 7): First equation: \(7 = 2(2) + 3 = 7\) ✓. Second equation: \(7 = -2 + 9 = 7\) ✓. Since (2, 7) satisfies both equations, it is the solution to the system. The other options fail to satisfy at least one equation.
Q2: How many solutions does the system \(y = 3x - 5\) and \(y = 3x + 2\) have? (a) No solution (b) Exactly one solution (c) Exactly two solutions (d) Infinitely many solutions
Solution:
Ans: (a) Explanation: Both equations have the same slope of 3 but different y-intercepts (-5 and 2). This means the lines are parallel and will never intersect. Therefore, the system has no solution.
Q3: When solving the system \(x + y = 10\) and \(x - y = 4\) by elimination, what is the value of \(x\)? (a) 3 (b) 5 (c) 7 (d) 9
Solution:
Ans: (c) Explanation: Adding the two equations: \((x + y) + (x - y) = 10 + 4\) gives \(2x = 14\), so \(x = 7\). The elimination method eliminates the \(y\) variable by adding equations with opposite coefficients.
Q4: A system of equations is graphed, and the two lines intersect at the point (-2, 5). What does this point represent? (a) The x-intercept of both lines (b) The y-intercept of both lines (c) The solution to the system (d) The slope of both lines
Solution:
Ans: (c) Explanation: The point of intersection of two lines represents the ordered pair that satisfies both equations simultaneously. Therefore, (-2, 5) is the solution to the system of equations.
Q5: Which system of equations would be best solved using the substitution method? (a) \(2x + 3y = 12\) and \(4x + 6y = 24\) (b) \(y = 5x - 7\) and \(3x + 2y = 16\) (c) \(x + y = 8\) and \(x - y = 2\) (d) \(6x + 2y = 10\) and \(3x - 2y = 5\)
Solution:
Ans: (b) Explanation: The substitution method is most efficient when one equation is already solved for a variable. In option (b), the first equation \(y = 5x - 7\) is already solved for \(y\), making it easy to substitute into the second equation.
Q6: What is the solution to the system \(2x + y = 7\) and \(y = 3\)? (a) (1, 3) (b) (2, 3) (c) (3, 1) (d) (4, 3)
Solution:
Ans: (b) Explanation: Since \(y = 3\) is given, substitute this value into the first equation: \(2x + 3 = 7\). Solving for \(x\): \(2x = 4\), so \(x = 2\). The solution is (2, 3).
Q7: Which of the following systems has infinitely many solutions? (a) \(y = 2x + 1\) and \(y = 2x - 1\) (b) \(y = x + 3\) and \(2y = 2x + 6\) (c) \(y = 4x\) and \(y = -4x\) (d) \(x + y = 5\) and \(x + y = 8\)
Solution:
Ans: (b) Explanation: A system has infinitely many solutions when the equations represent the same line. In option (b), dividing the second equation by 2 gives \(y = x + 3\), which is identical to the first equation. Therefore, every point on the line is a solution.
Q8: What is the first step in solving the system \(3x + 4y = 10\) and \(x = 2y - 1\) using substitution? (a) Add the two equations together (b) Multiply the first equation by 2 (c) Substitute \(x = 2y - 1\) into the first equation (d) Solve the first equation for \(x\)
Solution:
Ans: (c) Explanation: The substitution method requires replacing a variable in one equation with its expression from the other equation. Since \(x = 2y - 1\) is already solved for \(x\), we substitute this into the first equation: \(3(2y - 1) + 4y = 10\).
Section B: Fill in the Blanks
Q9:A system of two linear equations that has exactly one solution is called a(n) __________ system.
Solution:
Ans: consistent independent Explanation: A consistent independent system has exactly one solution because the lines intersect at one point. The term "consistent" means at least one solution exists, and "independent" means the equations are not multiples of each other.
Q10:When two lines in a system of equations are parallel, the system has __________ solution(s).
Solution:
Ans: no Explanation:Parallel lines have the same slope but different y-intercepts, so they never intersect. A system with parallel lines is called inconsistent and has no solution.
Q11:The point where two lines intersect on a graph represents the __________ to the system of equations.
Solution:
Ans: solution Explanation: The point of intersection contains the x and y values that satisfy both equations simultaneously, making it the solution to the system.
Q12:In the elimination method, if adding two equations eliminates a variable, the coefficients of that variable must be __________ of each other.
Solution:
Ans: opposites Explanation: For a variable to be eliminated by addition, its coefficients must be opposites (additive inverses), such as 3 and -3. When added, they sum to zero, eliminating the variable.
Q13:A system of equations where both equations represent the same line has __________ solutions.
Solution:
Ans: infinitely many Explanation: When two equations are equivalent and represent the same line, every point on that line is a solution. This type of system is called consistent dependent and has infinitely many solutions.
Q14:The __________ method involves solving one equation for a variable and then replacing that variable in the other equation.
Solution:
Ans: substitution Explanation: The substitution method is a technique for solving systems where one equation is solved for a variable, and that expression is substituted into the other equation.
Section C: Word Problems
Q15:The sum of two numbers is 24, and their difference is 6. Find the two numbers.
Solution:
Ans: Let \(x\) be the larger number and \(y\) be the smaller number. System of equations: \(x + y = 24\) and \(x - y = 6\) Adding the two equations: \(2x = 30\), so \(x = 15\) Substituting into the first equation: \(15 + y = 24\), so \(y = 9\) Final Answer: The two numbers are 15 and 9.
Q16:A movie theater charges $12 for adult tickets and $7 for child tickets. On a certain day, the theater sold 150 tickets and collected $1,450 in total revenue. How many adult tickets were sold?
Solution:
Ans: Let \(a\) = number of adult tickets and \(c\) = number of child tickets. System of equations: \(a + c = 150\) and \(12a + 7c = 1450\) From the first equation: \(c = 150 - a\) Substituting into the second equation: \(12a + 7(150 - a) = 1450\) \(12a + 1050 - 7a = 1450\) \(5a = 400\) \(a = 80\) Final Answer: 80 adult tickets were sold.
Q17:Maria has $3.50 in dimes and quarters. She has a total of 20 coins. How many quarters does she have?
Solution:
Ans: Let \(d\) = number of dimes and \(q\) = number of quarters. System of equations: \(d + q = 20\) and \(0.10d + 0.25q = 3.50\) From the first equation: \(d = 20 - q\) Substituting into the second equation: \(0.10(20 - q) + 0.25q = 3.50\) \(2 - 0.10q + 0.25q = 3.50\) \(0.15q = 1.50\) \(q = 10\) Final Answer: Maria has 10 quarters.
Q18:Two cyclists start from the same point and travel in opposite directions. One cyclist travels at 15 mph and the other at 12 mph. After how many hours will they be 81 miles apart?
Solution:
Ans: Let \(t\) = time in hours. Distance = rate × time, so the total distance apart is: \(15t + 12t = 81\) \(27t = 81\) \(t = 3\) Final Answer: They will be 81 miles apart after 3 hours.
Q19:A school cafeteria sells pizza slices for $2.50 and juice boxes for $1.25. If a student spends $15 and buys a total of 9 items, how many pizza slices did the student buy?
Solution:
Ans: Let \(p\) = number of pizza slices and \(j\) = number of juice boxes. System of equations: \(p + j = 9\) and \(2.50p + 1.25j = 15\) From the first equation: \(j = 9 - p\) Substituting into the second equation: \(2.50p + 1.25(9 - p) = 15\) \(2.50p + 11.25 - 1.25p = 15\) \(1.25p = 3.75\) \(p = 3\) Final Answer: The student bought 3 pizza slices.
Q20:A rectangle has a perimeter of 40 cm. The length is 4 cm more than twice the width. Find the dimensions of the rectangle.
Solution:
Ans: Let \(l\) = length and \(w\) = width. System of equations: \(2l + 2w = 40\) and \(l = 2w + 4\) Substituting the second equation into the first: \(2(2w + 4) + 2w = 40\) \(4w + 8 + 2w = 40\) \(6w = 32\) \(w = \frac{32}{6} = \frac{16}{3}\) cm or approximately 5.33 cm \(l = 2(\frac{16}{3}) + 4 = \frac{32}{3} + 4 = \frac{44}{3}\) cm or approximately 14.67 cm Final Answer: The width is \(\frac{16}{3}\) cm (or 5.33 cm) and the length is \(\frac{44}{3}\) cm (or 14.67 cm).
The document Worksheet (with Solutions): Systems of Equations is a part of the Grade 9 Course Integrated Math 1.
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