3. Solve the following system of equations by
and check your answer
Solve the first equation for y
Transpose the 4x to the other side of the equation.
Then divide both sides by 3.
Do the same thing for the other equation.
Now graph both equations in the same picture.
The lines look very close to being parallel, but they aren't. The
slope of the first line is -4/3, and the slope of the second line is -3/2.
Since these two numbers are fairly close together, these lines look
like they are almost parallel, but since the slopes are not exactly
equal, the lines are not exactly parallel. The second line has a slightly
steeper slope than the first line, so they will meet if we extend the
graphs far enough to the right. For the first line, a slope of -4/3
means that you go down 4 units (up -4 units) when you go over 3
units. For the second line, a slope of -3/2 menas that you go down 3
units when you go over 2 units.
and the lines finally meet at the point (15, -19).
Check: Let x = 15 and y = -19
And in the second equation
So both answers check.
In this case while we are quite fortunate that the lines meet at
a point that has whole number coordinates, we have to go for quite a
way before the lines actually meet. In this case one of the
computational methods might work better
Solve one of the equations for one of the unknowns. In this case it
will be simplest to solve the first equation for y; first, because that is
the unknown for which we will have the fewest fractions when we
solve, and second, because we have already solved the first equation
for y when we graphed the two equations.
Substitute this in for y in the other equation
Clear fractions by multiplying both sides by 3.
The simplest; way to find y at this point is to substitute this
solution into the equation where we solved for y as a function of x.
Fortunately, this simplifies
and we get the same solution which we have previously shown to
Since the coefficients of neither of the variables match up, we
need to multiply the equations by suitable numbers. In this case,
multiply the first equation by 3 and the second equation by -4.
We get our solution for y very quickly with this method. If one
is using only this method, then one will not have either equation
solved for either unknown at this point, so we will have to substitute
this solution into one of our original equations. If we substitute it
into the first equation we get
The same solution we got before.
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