# Kinematics SUVAT Simultaneous Equation

I have the following question:

A car accelerates for four seconds at a constant rate. Over the four seconds it travels 104m. It covered 58m in the last two seconds. Find the speed at which the car was traveling before it started to accelerate and the rate of acceleration (in $ms^{-2}$).

So I looked at this and saw the following two sets of information:

$t = 4, S = 104$

$t = 2, S = 58$

With $u$ and $a$ as our unknowns, I thought this would be the correct situation to use a simaltaneous equation in conjunction with the following equation to figure this out:

$S = ut + \frac{1}{2}at^2$

Which using the values listed previously would yield the two following equations:

$104 = 4u + 8a$

$58 = 2u + 2a$

Solving these simultaneously gives:

$4a = -12$

$a = -3ms^{-2}$

Substituting this back in to one equation gives $u$:

$2u = 58 + 6 = 64$

$u = 32ms^{-1}$

However it is obvious that neither of these can be right, I got a deceleration from my calculation when it clearly states that the vehicle is accelerating. The correct answers from the answers section are that $a = 3$ and $u = 20$, how do you get to these answers?

In the last $2$ seconds we travelled $58$, so in the first $2$ seconds we travelled $46$. At time $t=2$ our net displacement $S$ was $46$.
Your second equation should therefore be $2u+2a=46$.