Lesson 13

Use the lines to decide whether each statement is true or false. Be prepared to explain your reasoning using the lines.

1. A solution to \(8=\text-x+10\) is 2.
2. A solution to \(2=2x+4\) is 8.
3. A solution to \(\text-x+10=2x+4\) is 8.
4. A solution to \(\text-x+10=2x+4\) is 2.
5. There are no values of \(x\) and \(y\) that make \(y=\text-x+10\) and \(y=2x+4\) true at the same time.

Here are three systems of equations graphed on a coordinate plane:

1. Match each figure to one of the systems of equations shown here.

   1. \(\begin{cases} y=3x+5\\ y=\text- 2x+20 \end{cases}\)

   2. \(\begin{cases} y=2x-10\\ y=4x-1 \end{cases}\)

   3. \(\begin{cases} y=0.5x+12\\ y=2x+27 \end{cases}\)

2. Find the solution to each system and then check that your solution is reasonable on the graph.
   • Notice that the sliders set the values of the coefficient and the constant term in each equation.
   • Change the sliders to the values of the coefficient and the constant term in the next pair of equations.
   • Click on the spot where the lines intersect and a labeled point should appear.

Your teacher will give you a page with 6 systems of equations.

1. Graph each system of equations by typing each pair of the equations in the applet, one at a time.

2. Describe what the graph of a system of equations looks like when it has . . .
   1. 1 solution
   2. 0 solutions
   3. infinitely many solutions

Use the applet to confirm your answer to question 2.

Sometimes it is easier to solve a system of equations without having to graph the equations and look for an intersection point. In general, whenever we are solving a system of equations written as

\(\displaystyle \begin{cases} y = \text{[some stuff]}\\ y = \text{[some other stuff]} \end{cases}\)

we know that we are looking for a pair of values \((x,y)\) that makes both equations true. In particular, we know that the value for \(y\) will be the same in both equations. That means that

\(\displaystyle \text{[some stuff]} = \text{[some other stuff]}\)

For example, look at this system of equations:

\(\displaystyle \begin{cases} y = 2x + 6 \\ y = \text-3x - 4 \end{cases}\)

Since the \(y\) value of the solution is the same in both equations, then we know \(\displaystyle 2x + 6 = \text-3x -4\)

We can solve this equation for \(x\):

\(\begin{align} 2x+6 &= \text-3x-4&& \\ 5x+6 &=\text-4\ &&\text{add \(3x\) to each side}\\ 5x &=\text-10\ &&\text{subtract 6 from each side}\\ x &=\text-2\ &&\text{divide each side by 5}\ \end{align}\)

But this is only half of what we are looking for: we know the value for \(x\), but we need the corresponding value for \(y\). Since both equations have the same \(y\) value, we can use either equation to find the \(y\)-value:

\(\displaystyle y = 2(\text-2) + 6\)

Or

\(\displaystyle y = \text-3(\text-2) -4\)

In general, a system of linear equations can have:

• No solutions. In this case, the lines that correspond to each equation never intersect.
• Exactly one solution. The lines that correspond to each equation intersect in exactly one point.
• An infinite number of solutions. The graphs of the two equations are the same line!