16 Questions
What is the first step in the substitution method for solving systems of linear equations in two variables?
Choose one of the equations and solve for one variable
What is the purpose of multiplying both equations by necessary multiples in the elimination method?
To make the coefficients of one variable the same
What type of real-world problem can be modeled using linear equations in two variables?
Mixture problems
What does the point of intersection of two lines on a coordinate plane represent?
The solution to the system of equations
What is the result if the two lines in a graphical representation of a system of linear equations are parallel?
There is no solution to the system
What is the purpose of graphing a system of linear equations in two variables?
To visualize the solution and check the results obtained through algebraic methods
What is the last step in both the substitution and elimination methods for solving systems of linear equations in two variables?
Substituting the value obtained for one variable back into one of the original equations
What type of problem can be modeled using linear equations in two variables, where the goal is to find the optimal mixture of different concentrations of a substance?
Mixture problems
In a business, a company sells two types of products, A and B, with profit margins of $5 and $3 per unit, respectively. If the company has a fixed cost of $100 and wants to make a profit of $500, how many units of each product should it sell to achieve this goal? Assume that the company sells x units of A and y units of B.
The company needs to solve a system of linear equations, where one equation represents the total profit (5x + 3y = 500 + 100) and the other equation represents the constraint on the production (x + y = ?). The solution to this system will give the required number of units of each product.
A car travels from city A to city B at an average speed of 60 km/h and returns at an average speed of 40 km/h. If the total travel time is 5 hours, how far is city B from city A? Use linear equations to model this problem and solve it.
Let the distance between city A and city B be x km. Then, the time taken to travel from A to B is x/60 hours, and the time taken to return is x/40 hours. Since the total travel time is 5 hours, we can set up the equation x/60 + x/40 = 5. Solving this equation, we get x = 120 km.
A manufacturer produces two products, X and Y, with profit margins of $10 and $15 per unit, respectively. The company has a limited production capacity of 2000 units per day and wants to maximize its profit. If the production of X requires 2 hours of machine time and Y requires 3 hours, how many units of each product should the company produce to maximize its profit?
Let the company produce x units of X and y units of Y. The profit function is P(x, y) = 10x + 15y, subject to the constraints 2x + 3y 2000 and x, y 0. The company needs to solve this linear programming problem to find the optimal solution.
Solve the system of linear equations using the substitution method: 2x + 3y = 7 and x - 2y = -3.
Solve the second equation for x, which gives x = -3 + 2y. Substitute this expression into the first equation, which gives 2(-3 + 2y) + 3y = 7. Simplify and solve for y, which gives y = 2. Substitute y into one of the original equations to find x, which gives x = 1.
A graph of a linear equation in two variables is shown below. What is the equation of the line and what does it represent in the context of a business problem?graph of a linear equation
The equation of the line is y = 2x + 1. In the context of a business problem, this equation could represent the cost of producing x units of a product, where y is the total cost. The slope of 2 represents the cost per unit, and the y-intercept of 1 represents the fixed cost.
Solve the system of linear equations using the graphical method: x + 2y = 4 and 2x - 3y = -5.
Graph the two lines on a coordinate plane. The point of intersection represents the solution to the system, which is x = 1 and y = 3/2.
A company sells two products, A and B, with prices of $20 and $30 per unit, respectively. The company wants to find the point of intersection of the two lines representing the revenue equations, where the x-axis represents the number of units of A sold and the y-axis represents the number of units of B sold. What is the equation of the line representing the revenue equation for product A?
The revenue equation for product A is R = 20x, where x is the number of units of A sold. This equation represents a straight line passing through the origin with a slope of 20.
A system of linear equations has a unique solution, and the graph of the two lines is shown below. What can be concluded about the system?graph of a system of linear equations
The system has a unique solution, which means that the lines intersect at a single point. This implies that the system is consistent and has a single solution.
Study Notes
Linear Equations in Two Variables
Substitution Method
- A method for solving systems of linear equations in two variables
- Steps:
- Choose one of the equations and solve for one variable (e.g., x)
- Substitute the expression obtained in step 1 into the other equation
- Solve the resulting equation for the other variable (e.g., y)
- Substitute the value obtained in step 3 back into one of the original equations to find the value of the first variable (e.g., x)
Elimination Method
- A method for solving systems of linear equations in two variables
- Steps:
- Multiply both equations by necessary multiples such that the coefficients of one variable (e.g., x) are the same
- Add or subtract the equations to eliminate one variable (e.g., x)
- Solve the resulting equation for the other variable (e.g., y)
- Substitute the value obtained in step 3 back into one of the original equations to find the value of the first variable (e.g., x)
Application Problems
- Linear equations in two variables can be used to model various real-world problems, such as:
- Cost and revenue analysis
- Distance and time problems
- Work and rate problems
- Mixture problems (e.g., mixing different concentrations of a substance)
- These problems often involve finding the point of intersection of two lines, which represents the solution to the system of equations
Graphical Representation
- A system of linear equations in two variables can be represented graphically as two lines on a coordinate plane
- The point of intersection of the two lines represents the solution to the system of equations
- If the lines are parallel, there is no solution to the system
- If the lines coincide, there are infinitely many solutions to the system
- Graphing can be used to visualize the solution and check the results obtained through algebraic methods
Solving Systems of Linear Equations in Two Variables
Substitution Method
- Solves systems of linear equations in two variables by substituting one variable into another equation
- Involves four steps: solving for one variable, substituting into the other equation, solving for the other variable, and substituting back to find the first variable
Elimination Method
- Solves systems of linear equations in two variables by eliminating one variable through addition or subtraction
- Involves four steps: multiplying equations by necessary multiples, adding or subtracting to eliminate one variable, solving for the other variable, and substituting back to find the first variable
Real-World Applications
- Linear equations in two variables model various real-world problems, including:
- Cost and revenue analysis
- Distance and time problems
- Work and rate problems
- Mixture problems (e.g., mixing different concentrations of a substance)
- These problems often involve finding the point of intersection of two lines, representing the solution to the system of equations
Graphical Representation
- Systems of linear equations in two variables are represented graphically as two lines on a coordinate plane
- The point of intersection of the two lines represents the solution to the system of equations
- Parallel lines indicate no solution, while coinciding lines indicate infinitely many solutions
- Graphing visualizes the solution and checks algebraic method results
Linear Equation in Two Variables
Application Problems
- Linear equations in two variables are used to model real-world problems, including:
- Cost and revenue analysis to determine profit and loss
- Distance and time problems to calculate speed and velocity
- Supply and demand analysis to predict market trends
- Optimization problems to find the most efficient solutions
Substitution Method
- The substitution method is used to solve systems of linear equations in two variables.
- The steps involved in the substitution method are:
- Solving one equation for one variable
- Substituting the expression into the other equation
- Simplifying the resulting equation to solve for the other variable
- Finding the value of the other variable by substituting the value back into one of the original equations
Graphical Representation
- A linear equation in two variables can be represented graphically on a coordinate plane.
- The graph of a linear equation is a straight line, where every point on the line represents a solution to the equation.
- The x-axis and y-axis represent the two variables, and the point of intersection of the line with the axes represents the values of the variables.
- Graphical representation is useful for:
- Visualizing the solution to a system of linear equations
- Identifying the number of solutions (one, none, or infinitely many)
- Determining the nature of the solutions (unique, inconsistent, or dependent)
Solve systems of linear equations in two variables using the substitution and elimination methods. Learn the steps to find the values of x and y.
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