3.03 Solving systems using elimination

Equation 1 is multiplied by 2 to produce the system: \begin{aligned} 8x+2y&=22 \\x-2y&=5\end{aligned} Determine whether or not the two systems are equivalent. Then decide whether or not the two systems result in the same solution.

Two systems are equivalent if they have exactly the same graphs. The two systems have the same solution if they intersecct at the exact same point.

The systems are equivalent since they share the same graphs. Since the systems are equivalent, they must also intersect at the same point.

These two systems are equivalent because the equations 4x+y=11 and 8x+2y=22 are on the exact same line. Multiplying the entire equation by a constant does not change the set of x and y values that make the equation true.

Equation 2 is replaced with the sum of Equation 2 and 3 times Equation 1 to produce the system: \begin{aligned} 4x+y &=11\\13x+y&=38 \end{aligned} Determine whether or not the two systems are equivalent. Then decide whether or not the two systems result in the same solution.

Two systems are equivalent if they have the exact same graphs. The systems have the same solution if they intersect at the same point.

The systems of equations are not equivalent because the graphs are not exactly the same. However, both systems do have the same solution, because they intersect at the same point:

We can also solve both systems using substitution to verify that the solutions are equal:

Begin by numbering the equations, so the solving process is easier to explain:

The solution to this system is (3,-1).

For the second system of equations:

The solution to this system is also (3,-1).

\begin{aligned} 9x+y &=62 \\ 5x+y&=38 \end{aligned}

Numbering the equations will help us describe our solving process:

The goal of the elimination method is to combine the two equations until one variable is eliminated.

In this problem, the y term will be eliminated if we multiply equation 2 by -1 and add it to equation 1.

Start by multiplying equation 2 by -1:

Then, add equation 1 and equation 2 together:

\begin{aligned} &&9x &+y = &62& \\ + &&-5x&-y=&-38& \\ \hline \\ &&4x&+0y=&24& \end{aligned}

Now that we know what x equals, we can substitute it back into either of the original equations:

The solution to this system is the ordered pair \left(6,8\right).

We could also consider the first two steps as "subtracting the equations". The steps would look like this instead:

\begin{aligned} &&9x &+y = &62& \\ - &&(5x&+y=&38&) \\ \hline \\ &&4x&+0y=&24& \end{aligned}

\begin{aligned} 2x+3y &= 19 \\ 4x-y&=10\end{aligned}

Numbering the equations will help us describe our solving process:

There are multiple combinations that can be used to solve this system:

• Multiply equation 1 by 2 and subtract it from equation 2
• Multiply equation 1 by -2 and add it to equation 2
• Multiply equation 2 by 3 and add it to equation 1

We only need to choose one approach.

Multiply the first equation by -2:

Then, add equation 1 and equation 2 together:

\begin{aligned} &&-4x &-6y = &-38& \\ + &&4x&-y=&10& \\ \hline \\ &&0x&-7y=&-28& \end{aligned}

Now that we know what y equals, we can substitute it back into either of the original equations:

The solution to this system is the ordered pair \left(3.5,4\right).

Remember, the solution of the system is unchanged when multiples of one equation are added to the other equation. That means we can substitute our solution for y into either of the original two equations of the system or the resulting equation from when we multiplied the first equation by -2.

Write a system of equations for this scenario, where x represents Verna's Chemistry test score, and y represents her English test score.

Since we know that Verna's Chemistry test score is higher than her English test score, we should express the difference of the two scores as x - y to get a positive result.

There are two other constraints based on typical test scores that we can apply to this problem:0\leq x \leq 100 and 0\leq y\leq 100.

Solve the system of equations to find her test scores.

Since the equations have the same coefficient of y with opposite sign, we can add the two equations to eliminate y and solve for x first.

\begin{aligned} &&x &+y = &128& \\ + &&x&-y=&16& \\ \hline \\ &&2x&+0y=&144& \end{aligned}

So Verna scored 72 on her Chemistry test and 56 on her English test.

In this case, the two equations also had the same coefficient of x with the same sign. So we could have instead approached this by subtracting the two equations and solving for y first.

Does the solution make sense in terms of the context? Explain your answer.

Yes. Assuming that the tests were out of 100, then 72 and 56 are both valid test scores to obtain.