Question

How do you solve the simultaneous equations $7a - 3b = 17$ and $2a + b = 16?$

Answer 1

Hint: A set of simultaneous equations, which is also known as a system of equations or an equation system, is a finite set of equations representing some mathematical expressions, situations.
A system of equations or an equation system having one or two variables consist of a common solution. A system of equations can be a system of linear equations or a system of nonlinear equations.
System of linear equations: A system of linear equations is a collection of one or more linear equations having the same set of variables. A solution to a linear equation is an assignment of values to the variables such that all the equations are simultaneously satisfied. to solve the system of linear equations use elimination method.
Elimination method: Multiply both the given equations by a non-zero constant to make the coefficients of any variable numerically equal, then add or subtract so that equal coefficients of one variable get eliminated. Now solve the equations in one variable and get the value. Substitute the value of one variable into another equation and solve the equation.

Complete step-by-step answer:
Step: 1 The given equations are,
$
  7a - 3b = 17 \ldots \ldots \ldots \ldots \left( 1 \right) \\
  2a + b = 16 \ldots \ldots \ldots \ldots \left( 2 \right) \\
 $
Step: 2 Multiply the first equation by $\left( 1 \right)$.
$\left( {7a - 3b} \right) \times 1 = \left( {1 \times 17} \right) \ldots \ldots \ldots \left( 1 \right)$
Multiply the second equation by 3.
$\left( {2a + b} \right) \times 3 = \left( {3 \times 8} \right) \ldots \ldots \ldots \left( 2 \right)$
Step: 3
Add the both equations two eliminate one variable and solve,
$
  7a - 3b = 17 \ldots \ldots \ldots \ldots \left( 3 \right) \\
  \underline {6a + 3b} = 48 \ldots \ldots \ldots \ldots \left( 4 \right) \\
  13a = 65 \\
 $
Divide the equation both side with 13,
$
  \dfrac{{13a}}{{13}} = \dfrac{{65}}{{13}} \ldots \ldots \ldots \ldots \ldots \left( 3 \right) \\
   \Rightarrow a = 5 \\
 $
Step: 4 Substitute the value of $\left( {a = 5} \right)$in the first equation.
$
  7a - 3b = 17 \ldots \ldots \ldots \ldots \left( 1 \right) \\
   \Rightarrow 7 \times 5 - 3b = 17 \\
   \Rightarrow - 3b = 17 - 35 \\
   \Rightarrow 3b = 18 \\
 $
Divide both sides of the equation by 3.
$
  \dfrac{{3b}}{3} = \dfrac{{18}}{3} \\
   \Rightarrow b = 6 \\
 $
Therefore, the first variable$\left( {a = 5} \right)$and the second variable$\left( {b = 6} \right)$.

Therefore,
$
  a = 5 \\
  b = 6 \\
 $

Note:
Student should multiply the first equation by 1$\left( {7a - 3b} \right) \times 1 = \left( {1 \times 17} \right) \ldots \ldots \ldots \left( 1 \right)$and second equation by $\left( 3 \right)$$\left( {2a + b} \right) \times 3 = \left( {3 \times 8} \right) \ldots \ldots \ldots \left( 2 \right)$.
After performing the step one, they should add the both equations to eliminate the variables. They must know the elimination method to solve the system of equations involving the two variables.
They must apply the elimination method step by step, carefully to avoid any mistake which is done by most of the students.