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Questions and Answers
Solve the simultaneous equations: $2x + 3y = 12$ and $4x - y = 9$. What are the values of $x$ and $y$?
Solve the simultaneous equations: $2x + 3y = 12$ and $4x - y = 9$. What are the values of $x$ and $y$?
- (1, 3)
- (2, 2) (correct)
- (3, 2)
- (0, 4)
Which of the following is a solution to $3x + y = 7$ and $2x - y = 1$?
Which of the following is a solution to $3x + y = 7$ and $2x - y = 1$?
- (2, 3)
- (3, 2)
- (1, 4)
- (2, 1) (correct)
What is the value of $y$ if $x = 3$ in the equations: $x - y = 4$ and $2x + y = 7$?
What is the value of $y$ if $x = 3$ in the equations: $x - y = 4$ and $2x + y = 7$?
- 2
- -1
- -2
- 1 (correct)
Solve for $x$ and $y$: $x + 2y = 10$ and $3x - y = 5$. What are the values of $x$ and $y$?
Solve for $x$ and $y$: $x + 2y = 10$ and $3x - y = 5$. What are the values of $x$ and $y$?
What is the solution of the simultaneous equations: $y = 2x + 1$ and $4x - y = 7$?
What is the solution of the simultaneous equations: $y = 2x + 1$ and $4x - y = 7$?
Which of the following satisfies the inequality $2x - 5 < 3$?
Which of the following satisfies the inequality $2x - 5 < 3$?
Solve $3 - x \geq 5$. What is the solution?
Solve $3 - x \geq 5$. What is the solution?
What is the solution set for $4x + 1 > 9$?
What is the solution set for $4x + 1 > 9$?
Which of these values satisfies $-3x < 9$?
Which of these values satisfies $-3x < 9$?
What is the solution to the inequality $5x - 7 \leq 13$?
What is the solution to the inequality $5x - 7 \leq 13$?
Which of the following represents the region defined by $x \geq 0$ and $y \leq 2$?
Which of the following represents the region defined by $x \geq 0$ and $y \leq 2$?
The inequality $y > 3x - 1$ represents:
The inequality $y > 3x - 1$ represents:
What kind of line would $y = 2x + 3$ be in the coordinate plane?
What kind of line would $y = 2x + 3$ be in the coordinate plane?
If $x \leq 2$ and $y \geq 1$, which region is represented?
If $x \leq 2$ and $y \geq 1$, which region is represented?
The inequality $x - y \leq 4$ can be represented as:
The inequality $x - y \leq 4$ can be represented as:
If $x \geq 0$ and $y \geq 0$, what is the feasible region of $y \leq 4$ and $x + y \leq 6$?
If $x \geq 0$ and $y \geq 0$, what is the feasible region of $y \leq 4$ and $x + y \leq 6$?
Which is the objective function in linear programming?
Which is the objective function in linear programming?
What is the solution for maximizing $P = 3x + 2y$ subject to $x \geq 0, y \geq 0, x + 2y \leq 8, 2x + y \leq 10$?
What is the solution for maximizing $P = 3x + 2y$ subject to $x \geq 0, y \geq 0, x + 2y \leq 8, 2x + y \leq 10$?
Which of the following is not a feasible region condition?
Which of the following is not a feasible region condition?
If the constraints are $x \leq 5$ and $y \leq 3$, the feasible region is:
If the constraints are $x \leq 5$ and $y \leq 3$, the feasible region is:
Write $x^2 + 6x + 5$ in the form $(x + a)^2 - b$. What is the correct expression?
Write $x^2 + 6x + 5$ in the form $(x + a)^2 - b$. What is the correct expression?
What is the vertex form of $x^2 - 4x + 7$?
What is the vertex form of $x^2 - 4x + 7$?
Which of the following is equivalent to $x^2 + 8x + 16$?
Which of the following is equivalent to $x^2 + 8x + 16$?
Complete the square for $x^2 + 2x$. What is the resulting expression?
Complete the square for $x^2 + 2x$. What is the resulting expression?
Express $x^2 - 10x + 21$ in vertex form. What is the correct form?
Express $x^2 - 10x + 21$ in vertex form. What is the correct form?
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Study Notes
Simultaneous Linear Equations
- Solving simultaneous linear equations involves finding values of variables that satisfy all given equations.
- Substitution and elimination are common methods for solving.
Linear Inequalities
- Linear inequalities involve comparing linear expressions using symbols such as <, >, ≤, and ≥.
- Solve for the variable, remembering to reverse the inequality sign when multiplying or dividing by a negative number.
Regions in a Plane
- Linear inequalities can be graphically represented by shading the region satisfying the inequality.
- A solid line indicates the boundary is included, while a dotted line indicates it's excluded.
Linear Programming
- A method of optimization involving maximizing or minimizing a linear objective function subject to constraints.
- The feasible region is the area within the constraints, where solutions are valid.
Completing the Square
- A technique for rewriting quadratic expressions in the form (x + a)² + b.
- It helps identify the vertex of the parabola represented by the quadratic equation.
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