Equations on two grids
Look at the grids to the left. What $x$
makes them both have the same value (number of green squares
minus number of pink squares)?
| |
Notice that by answering this question, you found a solution to the
equation $x-3=1$. That is, the two grids to the left illustrate this equation.
Whenever you solve an equation, it’s a good idea to check your solution
by substituting it back into the original equation. Then make sure that
both sides of the result are equal.
Check your solution to this equation.
| |
Adding to both sides of an equation
Look at the two grids to the left, which currently illustrate the equation $x-5=2$. A second
slider has been added below the grids. This slider changes the equation by adding the same
number $k$ to both $x-5$ and $2$.
For each row of the table below, slide the $k$ slider to the value given in the
table. Then type in the equation that is illustrated by the two grids. Finally, use the $x$
slider to find the value of $x$ that solves that equation, as in the previous question.
If an $x$ value makes both sides of an equation equal, then the sides stay equal for that $x$
after you add some $k$ value to both sides.
After sliding the $k$ slider, you can simplify the equation you
get. For example, the
equation $x+2=6$ is currently pictured on the grids to the left. If you slide the $k$ slider
to $k=1$, you will get a picture of the equation $x+2+1=6+1$. But $x+2+1$ means the same thing
as $x+3$, and $6+1$ means the same thing as $7$. So a simpler way to write $x+2+1=6+1$ would be
$x+3=7$.
The equation $x+2=6$ is pictured on the grids to the left. Change this equation by
sliding the $k$ slider to each of the values in the table below, simplify the resulting
equation, and find the $x$ that solves it.
Does sliding the $k$ slider (that is, adding the same number to both
sides of the equation) change the value of $x$ that solves the equation?
(Type yes or no.)
| |
Notice that in the last row of the table, $x$ is alone on one side of the equation after
simplifying. That is, we have isolated $x$ from the other terms in the
equation.
Sliding the $k$ slider doesn’t change the solution to an equation. So you can slide $k$ to
some value that makes the equation simpler. If you slide $k$ to a value which
isolates $x$ — that is, puts $x$ alone on one side of the equation — you will have solved
the equation.
For example, the grid on the left shows the equation $x-1=4$. If you set $k=1$, the value of
the green square from the $k$ slider will cancel out the
value of the pink square from the original equation, so the top grid
will just have value $x$. That is, setting $k=1$ allows you to isolate $x$.
This means that you can solve the equation by adding 1 to both sides:
Click to see the equation $x-2=5$ pictured on the grids
to the left. Then, solve that equation by finding the $k$ value that isolates $x$.
Whenever you find the solution to an equation in this way, you should check
that it actually solves the equation.
Each row of the table below has an equation. Find the $k$ that isolates $x$ in
the equation. Then find and check the solution to the equation.
Notice that you always want $k$ to be the opposite of whatever number $x$ is being added to.
That is:
The equation $x+b=c$ can be made simpler by adding $-b$ to both sides of it
(or subtracting $b$ from both sides of it).
If you want to solve an equation and you aren’t given this kind of grid picture, you can do
the same thing. If the equation looks like $x+b=c$, you can solve it by:
- Adding $-b$ to both sides of the equation (or subtracting $b$ from both sides of the
equation).
- Simplifying the two sides of the equation until $x$ is isolated.
Solve each equation in this table by adding the appropriate number to both
sides and then simplifying. Then check your solution.