In this question we're asked,
what balances with one square shown here on the bottom?
And we're given that one circle and one square
balances with two triangles
and one circle and one triangle balances with two squares.
So hopefully you've already attempted this question.
If you haven't,
I hope you'll pause the video and give it a try.
Because we know one circle and one square
balances with two triangles,
let's begin by placing these on this first scale.
Again we'll place one circle and one square on the left
and two triangles on the right.
Now because we also know that one circle and one triangle
balances with two squares, notice how if we place one circle
and one triangle on the left and these two squares
on the right, the scale would also balance.
But instead of doing this, let's place these two squares
on the left and the circle and triangle on the right.
So we'll place two squares on the left
and a circle and triangle on the right.
Notice how when we do this we have one circle
on each side of the scale
and because the two circles have the same weight,
if we remove these, we now know that three squares
would balance with three triangles.
Well if three squares balance with three triangles
then one square would have to have the same weight
as one triangle
or one square would balance with one triangle.
Which is our answer.
Let's also try to solve this using algebra.
Using an algebraic approach,
let's let each circle be equal to x,
each square be equal to y,
and each triangle be equal to z.
And now we can form two equations
using these first two scales.
If a circle and a square balances with two triangles
that means that x plus y must equal two z.
And if one circle and one triangle balances with two squares
that would mean that x plus z must equal two y.
So notice how we have a system of equations here,
we have two equations with three unknowns
and our goal here is to determine
what would balance with one square or what y is equal to.
So to solve the system of equations for y let's solve
these two equations for x then we'll perform substitution.
So solving this first equation for x we would subtract
y on both sides which would give us x equals two z minus y.
We solve the second equation for x,
we would subtract z on both sides
so we'd have x equals two y minus z.
And now because we have x equals two z minus y
and x equals two y minus z,
we can form one equation with y and z,
we can substitute two z minus y for x here,
since two z minus y is equal to x.
So after performing this substitution
we'd have the equation two z minus y equals two y minus z.
And now to get the z terms on one side of the equation
and the y terms on the other,
let's go ahead and add y to both sides of the equation.
And let's also add z to both sides of the equation.
Now when we simplify, this would be zero and so would this,
so we're left with two z plus z which is three z
must equal two y plus y which would be three y.
So if we divide both sides by three,
notice how we have z equals y
or if we want y equals z and because y equals z
and y is a square and z is a triangle,
once again we know that one triangle
balances with one square.
I hope you found these explanations helpful.