Math Insight

Solving 2x2 linear systems

Math 2241, Spring 2023
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Due date: Feb. 1, 2023, 11:59 p.m.
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Total points: 1
  1. One way we use matrices is to express and solve systems of linear equations. Let's start by translating between a matrix equation and a system of linear equations.

    1. Let $A = \left[\begin{matrix}1 & 3\\-5 & 2\end{matrix}\right]$ and $\vc{x} = \left[\begin{matrix}x\\y\end{matrix}\right]$. Compute their product.
      $A\vc{x} = $

      Let $\vc{b} = \left[\begin{matrix}-1\\2\end{matrix}\right]$, then a matrix equation for $\vc{x}$ is $A\vc{x} = \vc{b}$. Both $A \vc{x}$ and $\vc{b}$ are two-dimensional vectors. Therefore, the equation $A\vc{x} =\vc{b}$ can be thought of as a system of two equations. The first equation is created by setting the first component of the vector $A\vc{x}$ equal to the first component of the vector $\vc{b}$. The second equation is created by setting their second components equal. Write down the system of two equations that is equivalent to the matrix equation $A\vc{x}=\vc{b}$.



      The idea is to solve the system equations for $x$ and $y$, which is the same as solving $A\vc{x}=\vc{b}$ for $\vc{x}$. But first, let's practice translating between matrix equations and their corresponding systems of equations.

    2. Write down the system of equations corresponding to the matrix equation $B\vc{x}=\vc{c}$ where \begin{align*} B = \left[\begin{matrix}2 & -1 & 3\\6 & 7 & -10\\-5 & 5 & 1\end{matrix}\right], \quad \vc{x} = \left[\begin{matrix}x\\y\\z\end{matrix}\right], \quad \vc{c} = \left[\begin{matrix}1\\2\\3\end{matrix}\right] \end{align*}




    3. Write down the matrix equation corresponding to the system of equations \begin{align*} 4 x + 2 y - 2 z &= -5\\ x + 5 y + 9 z &= 6\\ 3 x - y + 7 z &= -8 \end{align*}



      =

    4. The system of equations \begin{align*} x - y &= 4\\ x + y &= 3 \end{align*} in matrix form is $A \vc{x} = \vc{b}$ where
      $A = $
      and $\vc{b} = $
      .

  2. When we solve a linear equation of the form $A\vc{x}=\vc{b}$, we are determining what the vector $\vc{x}$ must be in terms of the matrix $A$ and the vector $\vc{b}$. There are many approaches to solving such systems of linear equations. We will focus on $2\times 2$ systems, but the method we will consider can be extended to systems of any size.

    For this problem, consider the system where $A=\left[\begin{matrix}4 & -2\\3 & 1\end{matrix}\right]$ and $\vc{b}=\left ( 10, \quad 5\right )$. If we think of $\vc{x}=(x,y)$, write down the system of equations corresponding to $A\vc{x}=\vc{b}$:



    1. What does it mean to solve a system of equations? Think about what it means to solve a single algebraic equation. For instance, to solve the equation $3x+1=7$, we need to find the value of $x$ such that the left-hand side of the equation equals
      when we plug $x$ in. With a system of equations, we want to find the values of all of the variables such that all of the equations are true.

      Since we know how to solve a linear equation with only one variable, one method for solving a system of linear equations is to eliminate one of the variables. Let's look at how to do this for our example.

      We'll try to eliminate $x$ first. One method to do so is to first multiply each equation by a constant so that the coefficients of $x$ in the two equations are the same. There are many ways to do this, but the simplest choice is to multiply each equation by the coefficient in the other equation. If we do this, the first equation should be multiplied by
      and the second equation should be multiplied by
      . The first equation is now


      and the second equation is now

      Now that the coefficients of $x$ match, we can eliminate $x$ by subtracting one equation from the other. We'll subtract the second equation from the first. What is the resulting equation?

      With $x$ eliminated, we can solve for $y$. We find that $y=$
      .

    2. The next step is to use $y$ to find $x$. We can use any of the equations to do this. We'll use the first equation (from above or from the beginning of the problem). We plug in the value of $y$ we found in part a to get an equation just involving $x$:

      We can now solve this for $x$, which tells us that $x=$
      .

      To summarize, the solution to the system of equations is $x=$
      , $y=$
      , which we can write in vector form as $\vc{x}=$
      .

  3. For this problem, consider the system \begin{eqnarray*} 5 x + 3 y&=&9\\ x - 3 y&=&9 \end{eqnarray*}

    1. We could follow the procedure outlined in the previous problem, but this particular problem lends itself particularly well to eliminating $y$ first. How are the coefficients of $y$ in the two equations related?
      Because of this, instead of multiplying each equation by some constant to get the coefficients to match and then subtracting, we can simply add to eliminate $y$. What is the resulting equation from adding the two equations together?

       $=$

      With $y$ eliminated, we can solve for $x$. We find that $x=$
      .

    2. The next step is to use $x$ to find $y$. We can use either equation to do this. We'll use the first equation. We plug in the value of $x$ we found in part a to get an equation just involving $y$:

       $=$

      We can now solve this for $y$, which tells us that $y=$
      .

      To summarize, the solution to the system of equations is $x=$
      , $y=$
      .

  4. Solve the following $2 \times 2$ systems.
    1. $A\vc{x} = \vc{b}$, where $$A=\left[\begin{matrix}2 & -5\\-1 & 4\end{matrix}\right], \quad \vc{b}=\left[\begin{matrix}-1\\5\end{matrix}\right]$$ $\vc{x} =$
    2. $B\vc{x}=\vc{c}$ , where $$B = \left[\begin{matrix}-5 & -3\\2 & 7\end{matrix}\right], \quad \vc{c} = \left ( 14, \quad 6\right ).$$ $\vc{x} = $
      Need help?
    3. \begin{eqnarray*} 3 x + 2 y &=& 1\\ 2 x - y &=& -11 \end{eqnarray*} $x=$

      $y=$

  5. In the multi-dimensional discrete systems worksheet, we introduced a dynamical system of bird populations on two islands, where each year there was reproduction and some migration between the islands. The result was that the dynamics of the population sizes $x_t$ and $y_t$ of islands one and two, respectively, in year $t$ were modeled by the discrete dynamical system: \begin{align*} x_{t+1} &= 0.88x_t + 0.24y_t \qquad \text{for $t=0,1,2,\ldots$}\\ y_{t+1} &= 0.22x_t + 0.96y_t. \end{align*} If in a given year there were $176$ birds on island one and $44$ birds on island two, how many birds must there have been on the island in the previous year?

    In the previous year, there must have been
    birds on island one and
    birds on island two.

    Need help?

  6. In all the above systems of two linear equations, how many solutions did you find?
    (Even though the solution might have two components, such $\vc{x}=(x,y)=(1,2)$, that's still considered one solution.)

    Sometimes, though, we can't solve the system of equations to get a unique value for $\vc{x}$.

    1. Try to solve the system of equations \begin{align*} 2x + 3y &= 10\\ 4x + 6y &= 12. \end{align*} Attempt to use the above algorithm to solve the equation before looking at the rest of this question.

      You should have run into a problem. If, for example, you eliminate $x$, what happens to $y$?
      Is the equation you get from eliminating $x$ true?
      Let's explore why.

      If $2x+3y = 10$, then what must $4x+6y$ be equal to?

      If $4x + 6y = 12$, then what must $2x+3y$ be equal to?

      Can you get $2x+3y$ to be $10$ and at the same time get $4x+6y$ to be $12$?
      For how many values of $\vc{x}=(x,y)$ is the system of equations satisfied?

      Need help?
    2. Try to solve the system of equations \begin{align*} 3x + 4y &= 5\\ -9x -12y &= -15. \end{align*} Attempt to use the above algorithm to solve the equation before looking at the rest of this question.

      You should have run into a problem. If, for example, you eliminate $x$, what happens to $y$?
      Is the equation you get from eliminating $x$ true?
      Let's explore why.

      If $3x + 4y = 5$, then what must $-9x -12y$ be equal to?

      If $-9x -12y = -15 $, then what must $3x + 4y$ be equal to?

      Can you get $3x + 4y$ to be $5$ without at the same time getting $-9x -12y$ to be $-15$?

      If we set $x=1$, find a value of $y$ so that the system of equations is solved. $y=$
      . Therefore, one solution is $\vc{x} = $

      If we set $x=3$, find a value of $y$ so that the system of equations is solved. $y=$
      . Therefore, one solution is $\vc{x} = $

      Are those the only two solutions to the system of equations?
      One could find a solution for
      value of $x$. Given a value of $x$, one just needs to set $y = $
      . For how many values of $\vc{x}=(x,y)$ is the system of equations satisfied?

      (Online, you can enter oo for $\infty$ or just use the symbol ∞.)

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Math 2241, Spring 2023