Question

Solve the simultaneous equations: \(2 x-3 y=11\) and \(5 x+4 y=16\) (1) \(x=5, y=-1 / 3\) (2) \(x=2, y=2 / 3\) (3) \(x=-1, y=4\) (4) \(x=4, y=-1\)

Short answer

Options: (1) \(x=5, y=-\frac{1}{3}\) (2) \(x=2, y=\frac{2}{3}\) (3) \(x=-1, y=4\) (4) \(x=4, y=-1\) Answer: (4) \(x=4, y=-1\)

Step-by-step solution

Set up the equations

Write down the given simultaneous equations: (1) \(2x - 3y = 11\) (2) \(5x + 4y = 16\)

Make the coefficients of x or y the same in both equations

To use the Elimination method effectively, we need to have the same coefficients for either x or y in both equations. We can achieve this by multiplying equation (1) by \(5\) and equation (2) by \(2\). This will give us the same coefficient of \(10\) for x in both equations. New equations after multiplication: (1) \(10x - 15y = 55\) (Multiply equation (1) by \(5\)) (2) \(10x + 8y = 32\) (Multiply equation (2) by \(2\))

Eliminate x

Subtract equation (2) from equation (1). By doing this, we eliminate the x terms. \((-15y - 8y) = (55 - 32)\) Simplify the equation: \(-23y = 23\)

Solve for y

Divide both sides of the equation by \(-23\): \(y = \frac{23}{-23}\) \(y = -1\)

Find the value of x

We will now use the value of y we found to solve for x. Substitute the value of y into either equation (1) or (2). We will use equation (1): \(2x - 3(-1) = 11\) Solve for x: \(2x + 3 = 11\) Subtract \(3\) from both sides: \(2x = 8\) Divide by \(2\): \(x = 4\)

Check the solution

Compare the values of x and y with the given options: (1) \(x=5, y=-\frac{1}{3}\) (2) \(x=2, y=\frac{2}{3}\) (3) \(x=-1, y=4\) (4) \(x=4, y=-1\) Our solution of \(x = 4\) and \(y = -1\) matches with option (4). Therefore, the correct pair of values for x and y is given by option (4).

Key concepts

Elimination Method

The Elimination Method is a popular technique for solving a system of equations. It involves combining equations to eliminate one of the variables, making it easier to solve for the remaining variable.

To start, you select a variable to eliminate. In our exercise, we chose to eliminate the variable \(x\). The key is to make the coefficients of this variable the same in both equations.

• For the given equations \(2x - 3y = 11\) and \(5x + 4y = 16\), multiply the first equation by \(5\) and the second by \(2\). This ensures the coefficient of \(x\) is \(10\) in both equations.

Once the coefficients match, subtract or add the equations to eliminate the chosen variable, leaving an equation with just one variable. Solving this simpler equation gives us a value for one variable, which leads us to the next concept: equation solving.

Equation Solving

Solving equations involves finding the value of the unknown variable(s) that satisfy the equation. After using the elimination method, we're left with a single equation to solve.

In our example, after eliminating \(x\), we got \(-23y = 23\). To solve for \(y\), you isolate the variable:

• Divide both sides of the equation by \(-23\).
• This gives us \(y = -1\).

With the value of \(y\) known, substitute it back into one of the original equations to solve for \(x\). This step-by-step substitution and solving ensure that we find both values that satisfy the entire system.

System of Equations

A system of equations contains two or more equations with the same set of unknowns. Simultaneous equations are a type of system where the goal is to find one solution that satisfies all equations.

In our problem, we have a system composed of:

• \(2x - 3y = 11\)
• \(5x + 4y = 16\)

Each equation represents a line on a coordinate plane. The solution to the system is the point where these lines intersect. Different methods, like the elimination method, give us the numerical values of this intersection point, solving the system. This leads us into considering the nature of these equations, known as linear equations.

Linear Equations

Linear equations are expressions that create straight lines when graphed. They take the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants and \(x\) and \(y\) are variables.

In the context of a system, each linear equation describes a line. The equations \(2x - 3y = 11\) and \(5x + 4y = 16\) both fit this model. These lines interact in space, and their overlap or intersection point(s) signifies the solution(s) to the system.

• One intersection point indicates a unique solution.
• No intersection points represent parallel lines (no solution).
• Infinite intersection points occur when the equations describe the same line (many solutions).

Understanding linear equations lays the foundation for grasping more complex systems and solution methods.