Linear Equations in Two Variables

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Questions and Answers

What is the first step in the substitution method for solving systems of linear equations in two variables?

Substitute the expression obtained for one variable into the other equation

Add or subtract the equations to eliminate one variable

Multiply both equations by necessary multiples

Choose one of the equations and solve for one variable (correct)

What is the purpose of multiplying both equations by necessary multiples in the elimination method?

To make the coefficients of one variable the same (correct)

To solve for one variable

To find the point of intersection

To eliminate one variable

What type of real-world problem can be modeled using linear equations in two variables?

Quadratic problems

Mixture problems (correct)

Trigonometric problems

Exponential growth problems

What does the point of intersection of two lines on a coordinate plane represent?

The solution to the system of equations Signup and view all the answers

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 Signup and view all the answers

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 Signup and view all the answers

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 Signup and view all the answers

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 Signup and view all the answers

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. Signup and view all the answers

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. Signup and view all the answers

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. Signup and view all the answers

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. Signup and view all the answers

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. Signup and view all the answers

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. Signup and view all the answers

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. Signup and view all the answers

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. Signup and view all the answers

Study Notes