Test for Substitution

Substitution is a method used to solve a system of linear equations in two variables. It involves solving one equation for one variable and substituting that expression into the other equation. This method is particularly useful when one of the equations is already solved for one variable.

To test the substitution method, let's consider the following system of equations:

Equation 1: 3x + 2y = 10
Equation 2: 4x - 5y = -8

Step 1: Solve one equation for one variable
In this step, we can choose either equation to solve for one variable. Let's solve Equation 1 for x:

3x = 10 - 2y
x = (10 - 2y) / 3

Step 2: Substitute the expression into the other equation
Now, we substitute the expression for x in Equation 2:

4((10 - 2y) / 3) - 5y = -8

Simplify the equation:

(40 - 8y) / 3 - 5y = -8

Step 3: Solve the equation
Next, we solve the equation obtained in Step 2. Multiply through by 3 to eliminate the fraction:

40 - 8y - 15y = -24

Combine like terms:

-23y = -64

Divide both sides by -23 to solve for y:

y = 64 / 23

Step 4: Substitute the value of y into one of the original equations
Now that we have found the value of y, we substitute it back into Equation 1 or Equation 2 to find the value of x. Let's substitute it into Equation 1:

3x + 2(64/23) = 10

Simplify and solve for x:

3x + 128/23 = 10
3x = 10 - 128/23
3x = (230 - 128) / 23
3x = 102 / 23
x = 34 / 23

Step 5: Check the solution
Finally, we check our solution by substituting the values of x and y into both original equations:

Equation 1: 3(34/23) + 2(64/23) = 10
Equation 2: 4(34/23) - 5(64/23) = -8

If the values satisfy both equations, then our solution is correct.

In conclusion, the substitution method is a systematic approach to solving a system of linear equations. By solving one equation for one variable and substituting the expression into the other equation, we can find the values of the variables.

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