Question

Solve the system of equations \[y = - 6x - 13\] and \[7x + 2y = - 16\]?

Answer 1

Hint: Use the combination method to solve two linear equations given in the question. We multiply both the equations separately with such constant values so that one of the variable values gets cancelled while adding or subtracting the two equations.
Combination method of solving systems of linear equations: We multiply the equation (or equations) with such constant values that make one of the values exactly equal in both linear equations. According to the signs of that value we add or subtract the equations.

Complete step by step solution:
We have two linear equations\[y = - 6x - 13\]
Shift 6x to left hand side of the equation
\[6x + y = - 13\] … (1)
And \[7x + 2y = - 16\] … (2)
We multiply equation (1) with constant 2
\[ \Rightarrow 12x + 2y = - 26\] … (3)
Now we subtract equation (3) from equation (2)
\[
  12x + 2y = - 26 \\
  \underline {7x + 2y = - 16} \\
  5x + 0y = - 10 \\
 \]
Now we get the equation as \[5x = - 10\]
Cancel same factors from both sides of the equation
\[ \Rightarrow x = - 2\] … (4)
Substitute this value of x back in equation (1) to calculate value of y
\[ \Rightarrow y = - 6( - 2) - 13\]
\[ \Rightarrow y = 12 - 13\]
\[ \Rightarrow y = - 1\]
So, value of x is -2 and value of y is -1
\[\therefore \]Solution of the system of linear equations \[y = - 6x - 13\] and \[7x + 2y = - 16\]is \[x = - 2;y = - 1\]

Note: Don't get confused between combination method and substituting method but they can remember the combination method by linking it to ‘linear combination’ which means one equation is linear combination of other i.e. we can convert one term of an equation with same variable equal to term of same variable of other equation. Also, always remember to change sign from positive to negative and vice-versa when subtracting equations.