What are simultaneous equations?

How can you solve the simultaneous equations x + y = 10 and x - y = 2?

Hint: You can use the substitution or elimination method.

Using the elimination method, we can add the two equations: (x + y) + (x - y) = 10 + 2, which simplifies to 2x = 12. Thus, x = 6. Substitute x back into one of the original equations to find y: 6 + y = 10, so y = 4. The solution is x = 6, y = 4.

What is the elimination method for solving simultaneous equations?

The elimination method involves adding or subtracting equations to eliminate one variable, allowing you to solve for the other variable. For example, if you have 2x + 3y = 6 and 4x - 3y = 12, you can add the two equations to eliminate y.

Solve the simultaneous equations 3x + 2y = 16 and x - y = 1.

Hint: Use substitution or elimination.

If x + y = 5 and x - y = 3, what are the values of x and y?

Hint: Use elimination.

Add the two equations: (x + y) + (x - y) = 5 + 3, which simplifies to 2x = 8. Thus, x = 4. Substitute x back into one of the equations: 4 + y = 5, so y = 1. The solution is x = 4, y = 1.

What is a key characteristic of a consistent system of simultaneous equations?

A consistent system of simultaneous equations has at least one solution, meaning the graphs of the equations intersect at one point or are identical (infinitely many solutions).

What is the graphical interpretation of simultaneous equations?

Simultaneous equations can be represented graphically as lines on a coordinate plane. The solution is the point where the lines intersect, representing the values of the variables that satisfy all equations.

Solve the equations 2x + 3y = 12 and x - y = 2.

Hint: Try substitution.

From the second equation, x = y + 2. Substitute into the first equation: 2(y + 2) + 3y = 12. This simplifies to 2y + 4 + 3y = 12, leading to 5y = 8, thus y = 1.6. Substitute back to find x: x = 1.6 + 2 = 3.6. The solution is x = 3.6, y = 1.6.

What do you call a system of equations with no solution?

A system of equations with no solution is called inconsistent. This occurs when the graphs of the equations are parallel and never intersect.

If 5x + 2y = 20 and 10x + 4y = 40, what can be said about the system?

Hint: Analyze the equations.

The second equation is just a multiple of the first (multiply the first by 2). This means they represent the same line, indicating that the system has infinitely many solutions.

How do you identify dependent equations in a system?

Dependent equations are equations that represent the same line. They will have the same slope and y-intercept, leading to infinitely many solutions.

What is the first step in solving simultaneous equations using the substitution method?

The first step in the substitution method is to solve one of the equations for one variable in terms of the other variable.

Solve for x and y given 4x - 5y = 10 and 2x + y = 7.

Hint: Use elimination or substitution.

What is the determinant in the context of a system of equations?

The determinant is a value calculated from the coefficients of a system of linear equations. It helps determine whether the system has a unique solution (determinant ≠ 0) or no solution (determinant = 0).

How do you solve a system of equations with three variables?

Hint: Use elimination or substitution methods iteratively.

To solve a system with three variables, you can use the elimination or substitution methods in combination. First, eliminate one variable between two equations, then use the resulting two-variable equations to find the other variables.

If x + y = 10 and 2x + 2y = 20, what type of system is this?

Hint: Consider the relationship between the two equations.

This system is dependent because the second equation is a multiple of the first equation, indicating that they represent the same line and have infinitely many solutions.

How do you check if your solution to simultaneous equations is correct?

To check if your solution is correct, substitute the values of the variables back into the original equations. If both equations are satisfied, your solution is correct.

What is a common mistake when solving simultaneous equations?

Hint: Think about combining equations.

A common mistake is incorrectly combining equations, such as adding or subtracting them without properly aligning the variables. This can lead to incorrect results.