Example 6
Solve the system of equations using elimination:
\(\begin{align*} 2x-100&=-9y \tag{1}\\ -4y-35&=-15x \tag{2} \end{align*}\)
Solution
Before we start with solving using elimination, we will first rearrange the equations so they are both in the form \(Ax+By=C\). This will allow us to easily compare the coefficients of \(x\) and \(y\) between the two equations.
\(\begin{align*} 2x+9y&=100 \tag{1}\\ 15x-4y&=35 \tag{2} \end{align*}\)
Step 1:
In this example, we will need to multiply by a factor to get the same or opposite coefficient for \(x\) or \(y\). If we wish to not use factors that are fractions we will need to multiply both equations by their own factor.
In our example, we could:
- choose to multiply so both equations have a coefficient of \(30\) (the lowest common multiple of \(2\) and \(15\)) in front of \(x\) \(\rightarrow\) Equation \((1) \times 15\) and Equation \((2) \times 2\)
or we could
- choose to multiply so both equations have a coefficient of \(36\) but with opposite signs (the lowest common multiple of \(9\) and \(4\)) in front of \(y\rightarrow\) Equation \((1)\times 4\) and Equation \((2)\times9\)
For our solution, we will choose the second option. This will give us two equations with opposite coefficients in front of their \(y\) variable.
- Equation \((1) \times 4\rightarrow \) Equation \((3)\)
- Equation \((2) \times 9\rightarrow\) Equation \((4)\)
\(\begin{align*} 8x+36y&=400 \tag{3}\\ 135x-36y&=315 \tag{4} \end{align*}\)
Steps 2 and 3:
We now have opposite coefficients for \(y\) so we will add the equations together to eliminate \(y\) and then we can solve for \(x\).
- Equation \((3)\) \(+\) Equation \((4)\):
\(\begin{align*} 8x+36y&=400 \tag{3}\\ 135x-36y&=315 \tag{4} \\ \hline 143x\;\;\;\;\;\;\;\;\;\;&=715 \tag{3)+(4}\\ x&=5 \end{align*}\)
Step 4:
Substitute \(x=5\) into Equation \((1)\).
\[\begin{align*} 2x+9y&=100\\ 2(5)+9y&=100\\ 9y&=100-10\\ 9y&=90\\ y&=10 \end{align*}\]
Step 5:
Verify the solution \((x,y)=(5,10)\) in Equation \((2)\).
\[\begin{align*}
\text{LS}&=15x-4y\\
&=15(5)-4(10)\\
&=75-40\\
&=35
\end{align*}\]
\[\text{LS}=\text{RS}\]
Therefore, \(\text{LS}=\text{RS}\) and \((5,10)\) is the solution to the system of equations.