Solving Equations

Basic Skills

Distribute.

Distributing involves applying a leading coefficient or sign to all of the terms within a pair of parentheses or other grouping of numbers/variables. For example, when a -5 is put in front of the binomial (x + b), it affects both of those terms, even though when it is written out ( − 5(x+b)) the coefficient is located next to the x variable and is separated from the b constant.

This means that when a student “distributes”  − 5(x+b) =  − 5x  − b. It’s important to note that not only the number is distributed, but also the sign. If both the sign on the coefficient and the sign in between the binomial terms are positive, then no signs will change on the terms. If the signs of the coefficient and between the binomial terms are either different or both negative, then a sign change will occur. As soon as a student sees a negative sign or a minus in between binomial terms, they should double check that they’ve distributed the signs correctly. 

Combine like terms.

If there are two or more similar terms on the same side of an equation, those terms can be combined. “Like” or “similar” terms means terms with the same exponent and the same variable. This is why a student can combine 3x + 5x but not 3x + 5x².

For example, if an equation looks like: 3x + 17 + 2x  − 5= 3x. On the left side of the equation there are two constant terms (terms that are just whole numbers) and two x terms (same variable and same exponent): these pairs can each be combined to simplify the equation.

This is how 3x + 17 + 2x  − 5= 3x becomes 5x  + 12  = 3x

Special Skills

Creating Equations from Word Problems

Some ACT word problems are much easier to solve with equations, even though the initial question might not provide an equation. Here is an example from the July 2020 ACT

Clearly there are two things that the question wants students to keep track of: time and distance. The key information given by the equation are the two speeds, the directions (due west/east, meaning the drivers are moving horizontally), and the total distance traveled by the two drivers.

In this case, to make the math easier, it’s important to recognize that even though the question asks about the intersection of two different journeys, it’s not necessary to create two different equations. The two drivers will pass each other when they’ve driven a total of 240 miles between them.

The equation that can be created is: 240 = 57x + 68x or (after combining like terms)

240 = 135x. This is because the drivers cover 135 miles (combined) each hour.

x = 240125=1.92 This is the number of hours it will take for the two drivers to pass each other/drive the 240 miles.

Since the question wants a time in minutes, the hours need to be converted: 

1.92 hours ∙60 minuteshour=115.2 One hour and 55 minutes after the initial departure time = 3:55pm. The answer is D.