Question

Write a brief paragraph outlining your strategy for solving a system of two linear equations containing two variables.

Short answer

Identify the equations, choose substitution or elimination, then solve for each variable step-by-step and verify.

Step-by-step solution

- Identify the Equations

First, identify the two linear equations in the system. They will generally be in the form: \(a_1x + b_1y = c_1\) \(a_2x + b_2y = c_2\)

- Choose a Solution Method

Decide whether to use substitution or elimination to solve the system. Substitution involves solving one of the equations for one variable and then substituting that expression into the other equation. Elimination involves adding or subtracting the equations to eliminate one of the variables.

- Apply the Method

If using substitution, solve one equation for either x or y and substitute this expression into the other equation. If using elimination, multiply one or both equations by necessary values so that adding or subtracting them will eliminate one variable.

- Solve for One Variable

After substitution or elimination, solve the resulting single-variable equation to find the value of one variable.

- Solve for the Other Variable

Use the value of the first variable to substitute back into one of the original equations and solve for the second variable.

- Verify the Solution

Finally, plug both values back into the original equations to ensure they satisfy both equations. This confirms that the solution is correct.

Key concepts

Substitution method

The substitution method is one way to solve a system of linear equations. It involves isolating one variable in one of the equations and then substituting that expression into the other equation. Here’s a step-by-step guide to help you understand and apply the substitution method:

• First, solve one of the equations for one of the variables. It doesn't matter which variable or which equation you choose. For example, if you have the equations:
  
  \(x + y = 5 \) and \(2x - y = 1\), you could solve the first equation for y:


• \( y = 5 - x \)


• Next, substitute this expression into the other equation. In the second equation \( 2x - y = 1 \) replace y with \( 5 - x \):
  \(2x - (5 - x) = 1 \)


• Simplify the equation and solve for x. In this case:
  \(2x - 5 + x = 1\)
  \(3x - 5 = 1\)
  \( 3x = 6\)
  \( x = 2 \)


• Finally, substitute the value of x back into the equation we used to express y:
  \( y = 5 - 2 \)
  \( y = 3 \)


• So, our solution for this system of equations is \( (x, y) = (2, 3) \).

This method simplifies the problem by reducing the number of variables you need to solve for at each step.

Elimination method

The elimination method is another technique for solving systems of linear equations. It involves combining the equations in a way that eliminates one of the variables. Here’s how you can use the elimination method effectively:

• First, write your system of linear equations. For example: \(x + y = 5\) and \(2x - y = 1\).


• Next, you need to align the equations so one of the variables can be canceled out. In this case, you can add the two equations directly because the y terms will cancel each other out:
  \((x + y) + (2x - y) = 5 + 1\)


• This simplifies to:
  \(3x = 6 \)
  Then, solve for x: \( x = 2 \)


• With the value of x known, substitute it back into one of the original equations to find y. For instance, using \( x + y = 5 \):
  \( 2 + y = 5 \)
  \( y = 3 \)


• Your solution is \( (x, y) = (2, 3) \).

This method is particularly useful when the coefficients of one of the variables are already opposite in sign or can easily be made so by multiplication.

Verification of solution

After solving a system of linear equations using either the substitution or elimination method, it’s crucial to verify that your solutions satisfy the original equations. This ensures that your solution is correct. Here's how to do it:

• Take the solution you found. Let's use our example where \( x = 2 \) and \( y = 3 \).


• Substitute these values back into the original equations. For the first equation: \( x + y = 5 \). Plugging in the values, we get \( 2 + 3 = 5 \), which is true.


• Now check the second equation: \( 2x - y = 1 \). Plugging in the values, we get \( 2(2) - 3 = 1 \), which also holds true.


• If your solutions satisfy both original equations, then your solution is correct.

Verifying the solution not only gives you confidence but also ensures you have a reliable result.