ACT Math - How to find the solution for a system of equations

Jacob is 3 years older than Sarah, and Caroline is twice as old as Sarah. If Caroline is 28 years old, how many years old is Jacob?

Explanation

One can describe the ages of Jacob, Sarah, and Caroline with the letters J, S, and C, respectively. From the information in the problem, J = S + 3, and C = 2S. Since C = 28, S = 28/2 = 14, and J = 14 + 3 = 17. Jacob is 17 years old.

Solve for x based on the following system of equations:

x + y = 5

2x + 3y = 20

_–_5

_–_10

_–_2

Explanation

One method of solving a system of equations requires multiplying one equation by a factor that will allow for the removal of one variable. In this system, we can multiply (x+y=5) by -2. When the -2 is distributed across the entire equation, the equation becomes (-2x-2y=-10). We then add the two equations: (-2x-2y=-10) + (2x+3y=20). When we do this, the x variable cancels out, leaving us with y=10. We then subsitute 10 for y in either of the original equations: (x+10=5) or (2x+ 30=20). Either way, we end up with x=-5. If you got an answer of 5, you may have made a computation error. If you got 10, you may have forgotten to substitute the y-value into one of the original equations to solve for x.

and

$y=\frac{t}{4}+5$

Which of the following expresses in terms of ?

$x=\frac{3y-63}{4}$

$x=\frac{19y-52}{6}$

$x=\frac{6y-48}{4}$

Explanation

First we must solve for , then substitute into the other equation. Since we want in terms of , solve for in the equation and substitute our value of (in terms of ) into the equation, then simplify:

$\frac{x+12}{3}=t$

Now that we have , let's plug that into the equation.

$y=\frac{\frac{x+12}{3}}{4}+5$

Already we can see that this problem is a mess because it is an expression with two denominators. Remember that dividing by a number is equal to multiplying by that number's inverse. Thus, dividing by is the same as multiplying by $\frac{1}{4}$ . So let's make an equivalent expression look like this:

$y=\frac{1}{4}\cdot \frac{x+12}{3}+5$

This is much better as we can multiply straight across to get:

$y=\frac{x+12}{12}+5$

Now we can solve for .

$y-5=\frac{x+12}{12}$

Find a solution to the system of equations:

2_x_ – y = 0

x + y = 3

(1,2)

(2,1)

(1,0)

(0,2)

(0,0)

Explanation

Use substitution and plug in to solve for one equation. Then use back substitution to solve for the other variable.

Given the following two equations, solve for :

$3a+2b=16$ $3a-2b=4$

$\frac{20}{6}$

$\frac{6}{20}$

Explanation

Solution A:

Notice that the two equations have very similar terms. If the two expressions are subtracted from each other, the variable cancels out:

$3a+2b=16$

$-(3a-2b=4)$ (don't forget to distribute the minus sign throughout!)

-----------------------

$4b=12$

$b=3$

Solution B:

Using one of the equations, solve for a in terms of b:

$3a+2b=16$

$3a=16-2b$

$a=\frac{16-2b}{3}$

$3(\frac{16-2b}{3})-2b=4$

$(16-2b)-2b=4$

$-4b=-12$

$b=3$

Note: Solution A is the much faster way to solve this problem. Whenever you are asked to solve a problem with two equations and two variables (or more!), see if you can add them together or subtract them from each other to make the other variables cancel out.

A store sells 17 coffee mugs for $169. Some of the mugs are $12 each and some are $7 each. How many $7 coffee mugs were sold?

Explanation

The answer is 7.

Write two independent equations that represent the problem.

x + y = 17 and 12_x_ + 7_y_ = 169

If we solve the first equation for x, we get x = 17 – y and we can plug this into the second equation.

12(17 – y) + 7_y_ = 169

204 – 12_y_ + 7_y_ =169

–5_y_ = –35

y = 7

Jenna's family owns a fruit stand. They began selling apples in 2010. In 2011 the number of apples sold increased by 100 apples. In 2012 they sold three times as many apples as they had in 2011, and in 2013 the number of apples increased by 200. If they sold 1700 apples in 2013, how many apples did they sell in 2011?

Explanation

In 2010 the family sold number of apples.

This increased by 100 in 2011: .

In 2012 the number of apples tripled: $3\left ( X + 100\right )$

In 2013 the number of apples increased by 200: $200 + \left [ 3 \left ( X + 100\right )\right ]$

If $200 + \left [ 3 \left ( X + 100\right )\right ]$ , we must solve for.

so in 2011 the number of apples sold was 500, or apples.

Suppose x2 + x – 6 = 0. Which of the following could be a value of x?

Explanation

Factor out this binomial. –3 and 2 are the only possible x values. 2 is the answer.

Sales for a business were $\textup{7 million dollars }$ more the second year than the first, and sales for the third year were double the sales for the second year. If sales for the third year were $\textup{46 million dollars}$ , what were sales, in millions of dollars, for the first year?

Explanation

The easiest way to solve this equation is to work backwards. Start with the $\textup{46 million}$ for the third year and divide by to get the second year's earning which were $\textup{23 million}$ , then subtract $\textup{7 million}$ to get the first year which was $\textup{16 million dollars}$ .