# 8-7 It's in the System - Concepts and Explanation

## Solving Linear Equations

Students have used tables or graphs to find solutions. They can solve simple linear
equations, *y* = *mx* + *b* or *mx* + *b* = *nx* + *c*, and simple equations with parentheses, *y* = *a*(*x* + *b*). In this Unit, students solve equations for different variables symbolically, writing
equivalent forms of the equation.

### Example

1. Subtract 12x from each side of the equation.

12*x* + 3*y* = 9

3*y* = -12*x* + 9

2. Divide each side of the equation by 3.

*y* = -4*x* + 3

1. Divide each side of the equation by 3.

12*x* + 3*y* = 9

Subtract 4*x* from each side of the equation

4*x* + *y* = 3*y* = 3 -4*x*

3. Rearrange the order of terms

*y* = -4*x* + 3

## Solving Linear Inequalities

Solving an inequality is very similar to solving a linear equation. The rules for operations with inequalities are identical to those for equations, with one exception. When multiplying (or dividing) an inequality by a negative number, you must reverse the direction of the inequality sign.

### Example

5*x *+ 7 ≤ 42

5*x* ≤ 35*x* ≤ 7

Solving this inequality is similar to solving 5*x* + 7 = 42. The operations (+, -, *x*, ÷) are applied to each side of the inequality. You usually show this solution on
a number line.

-5*x* + 7 ≤ 42

-5*x* ≤ 35*x* ≥ -7

Reverse the direction of the inequality sign.

## Solving Systems of Linear Equations

There are three standard methods for solving a system of linear equations.

The **graphing method** involves producing straight-line graphs for each equation and then reading coordinates
of intersection points as the solution(s).

The **linear combination method** relies on two basic principles; (1) If one of the equations is replaced by a new
equation formed by adding the two original equations, the solution is unchanged. (2)
The solutions of any linear equation A*x* + B*y* = *C* are the same as the solutions of *KAx* + *KBy* = *KC*, where *K* is a nonzero number.

The **equivalent form method** is the process of rewriting the equations in *y* = *ax* + *b* form and then setting the two expressions for *y* equal to each other.

### Example

The intersection point has coordinates (30, 20), so the solution of the system is
*x* = 30 and *y* = 20.

Adding the two equations gives -9*y* = -9. The solution is *y* = 1 and *x* = 1.

Since *y* = *y*, -2*x* + 5 = 3*x* - 5. The solution is *x* = 2 and *y* = 1.

## Solving Systems of Linear Inequalities

Systems of inequalities tend to have infinite solution sets. The solution of a system of distinct, nondisjoint linear inequalities is the intersection of two half-planes, which contain infinitely many points

### Example

In general, there are four regions suggested by a system of linear inequalities such
as the following:

Region 1 contains the solutions to the system. Points in Regions 2 and 3 satisfy one, but not both, of the inequalities. The fourth region satisfies neither inequality.