# ACT Math Skill Review: Linear Equations

You will sometimes be asked to solve systems of two or more linear equations or inequalities. Linear systems are equations that contain the same variables. For example, *a* + *b* = 11 and 2*a* + *b* = 10 are considered linear systems since both contain the same variables, *a* and *b*.

To solve systems of linear equations, you can use the substitution method:

- Take one of the equations listed and find the value of one of the variables in terms of the other.
- Substitute the value found for the variable into the other equation.
- Solve for the second variable.
- Substitute that value into the original equation to solve for the first variable.

Let’s see how this works:

For what values of *x* and *y* are the following equations true?

In the first equation, solve for *x* in terms of *y*.

Then substitute this value for *x* into the second equation and solve for *y*.

Substitute *y* = 9 back into the original equation to find the value of *x*.

#### Examples

Try the following examples on your own:

#### Answers and Explanations

**The correct answer is A.**You want to solve for*x*, so let’s start by putting*y*in terms of*x*, using the second equation:*y*= 4 – 2*x*. Substitute that value for*y*back into the first equation: 3*x*+ 5(4 – 2*x*) = 6. This becomes 3*x*+ 20 – 10*x*= 6, and then -7*x*= -14. Thus,*x*= 2.

**The correct answer is J.**If you have 3 variables, you need 3 equations. It’s the same process as with two equations, you just have a few extra steps. Start with the third equation and put*x*in terms of*y*and*z*. You end up with 2*x*=*z*–*y*. Stick with 2*x*as it’ll be easier to work with when substituting it back into the other equations. Take this value and substitute it into the second equation for x. You end up with 2(*z*-*y*) +*y*– 3*z*= -6. This becomes 2*z*– 2*y*+*y*– 3*z*= -6. From there, isolate*y*to get*y*in terms of*z*: -*z*-*y*= -6,or*y*= 6 -*z*. Now plug this value for*y*into the value for 2*x*, to get*x*in terms of*z*: 2*x*=*z*– (6 -*z*). This becomes 2*x*= 2*z*– 6. You now have values for both 2*x*and*y*in terms of*z*. Substitute these values into the first equation to get: 4(2*z*– 6) + 3(6 –*z*) + 2*z*= 22. Continue to simplify: 8*z*– 24 + 18 – 3*z*+ 2*z*= 22 and 7*z*– 6 = 22. Finally, 7*z*= 28 and*z*= 4.

- Time Management
- Number Properties
- Fractions
- Arithmetic
- Order of Operations
- Algebraic Manipulation
- Word Problems
- Solving Equations
- Inequalities
- Linear Equations
- Miscellaneous Algebra
- Functions
- Ratios, Proportions and Variations
- Lines & Angles
- Triangles
- Quadrilaterals
- Circles
- Graphs and Coordinate Geometry
- Probability and Statistics
- Trigonometry

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