Simultaneous Linear Equations in Two Variables

6.3

Simultaneous Linear Equations in Two Variables

Example

Equations: \(x+y=7\), \(2x+y=12\)

Both equations are simultaneous linear equations in two variables because both linear equations have two similar variables.

Solve simultaneous linear equations in two variables:

Graphical method:

Example

Solve: \(x + y = 6\) and \(2x + y = 8\)

From the graph above, the point of intersection is \((2,4) \).

Thus, the solution is \(x=2\) and \(y=4\).

Substitution method:

Express one of the variables in terms of the other variable.
Substitute the expression into the other linear equation.
Solve the linear equation in one variable.
Substitute the value obtained into the expressed equation to find the value of the other variable.

Example

Solve: \(x – 3y = 7\) and \(5x + 2y = 1\)

\(\begin{aligned} x-3y&=7....\boxed{1} \\\\5x+2y&=1....\boxed{2} \end{aligned}\)

From \(\boxed{1}\), \(x=7+3y....\boxed{3}\).

Substitute \(\boxed{3}\) into \(\boxed{2}\),

\(\begin{aligned} 5(7+3y)+2y&=1 \\\\35+15y+2y&=1 \\\\35+17y&=1 \\\\17y&=1-35 \\\\17y&=-34 \\\\y&=-2. \end{aligned}\)

As \(y=-2\),

\(\begin{aligned} x&=7+3(-2) \\\\&=1. \end{aligned}\)

Elimination method:

Multiply one or both equations with a number so that the coefficient of one of the variables is equal.
Add or subtract both the equations to eliminate one of the variables.
Solve the linear equation in one variable.
Substitute the value obtained into the original equation to find the value of the other variable.

Example

Solve: \(x + 2y = 9\) and \(3x – 2y = 15\)

\(\begin{aligned} x+2y&=9....\boxed{1} \\\\3x-2y&=15....\boxed{2} \end{aligned}\)

Eliminate the variable \(y\) by adding \(\boxed{1}\) and \(\boxed{2}\).

\(\begin{aligned} x+2y&=9 \\\\+\quad3x-2y&=15 \\\underline{\hspace{2cm}}&\underline{\hspace{2cm}} \\4x+0&=24 \\\\4x&=24 \\\\x&=6. \end{aligned}\)

Substitute \(x=6\) into \(\boxed{1}\),

\(\begin{aligned} 6+2y&=9 \\\\2y&=9-6 \\\\2y&=3 \\\\y&=\dfrac{3}{2}. \end{aligned}\)