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# Solving
Equations

### Solving
Linear Equations

A linear equation has
the form ax + b = c, where a, and b are constants and x is the variable.
To solve a linear equation, we must find the value of the variable
which make the equation true. An algebraic method that can be used
to solve linear equations is the following: Replace the equation
with a simpler equivalent one and continue this process until a
solution is reached. Below are some procedures that result in equivalent
equations.

Interchange the two
sides of the equation.

Simplify either or
both sides of the equation by removing parenthesis, adding like
terms, etc.

Add or subtract the
same expression from both sides of the equation.

Multiply or divide
both sides of the equation by the same non-zero expression.

The following examples
illustrate these ideas.

### Example

Solve the linear equation
100 = -7.2t + 250 for t. Round the answer to two decimal places
if necessary.

### Solution

_{}

### Example

Solve the linear equation
–3.2(0.7x + 12) +10.5 = -0.8x + 1.6. Round the answer to two decimal
places.

### Solution

_{} _{}

## Solving
Quadratic Equations

### Factoring Method

A quadratic equation
has the form

_{}

If the expression _{} can be factored, then we can set each factor
equal to zero. The following example illustrates the factoring method.

### Example

Solve the quadratic equation

_{}.

### Solution

### The Quadratic Formula

Since most
quadratic expressions cannot be factored, the factoring method is
limited in its usefulness. However, any quadratic equation
_{} that has real solutions can be solved
by the quadratic formula

_{}

The quantity _{} is called the discriminant of the quadratic
equation because it determines whether the equation has one, two,
or no real solutions:

If _{}, the equation has two real solutions.

If _{}, the equation has exactly one solution.

If _{}, the equation has no real solutions.

### Example

Use the quadratic formula
to find the solutions, if any, of the quadratic equation _{}

### Solution

Comparing our equation
to _{}, we see that _{} Substituting these values into the quadratic
formula we get

_{}

Evaluating this
expression, we find two solutions, x = -5.49 and x =1.17 (rounded
to two decimal places).

### Example

Use the quadratic formula
to find the solutions, if any, of the equation _{}

### Solution

In this case, we must
first subtract 160 from both sides so that the equation is in the
form _{}.

_{}

We now see that _{} Substituting these values into the
quadratic formula, we get,

_{}.

Evaluating this expression
yields the two solutions t= -0.71 and t = 3.52 (rounded to two decimal
places).

## Solving
Equations Graphically

Almost any equation in
one variable can be solved graphically. The strategy is as follows:
graph the left hand side and the right hand side of the equation
on the same coordinate axes, and then trace the graph to find the
point(s) where the two curves intersect. If there is no point of
intersection then we say that the equation has no solution. The
following example will illustrate this procedure.

Example

Find all solutions of
the equation _{}. Round your answers to three decimal places.

Solution

First we will use the
Plot-Solve tool to graph the left-hand side of the equation, _{}and the right-hand side, _{} on the same coordinate axes. See Figure 1.

Figure 1

The graph shows that
there are two points of intersection. Click on one of the points
and zoom in to find the coordinates of that point of intersection.
Figure 2 shows that the intersection point on the left has coordinates
(-3.3, 60).

Figure 2

We can repeat the process
to find the right intersection point. The coordinates are

(45.2, 60). See Figure
3.

Figure 3

The equation has two
solutions, _{} (accurate
to one decimal place).