Elementary Algebra Review

SYSTEMS OF EQUATIONS

Solving Systems of Equations by Graphing

A system of equations is two or more equations considered together. A system in two variables can be
solved by graphing the two lines on the same coordinate axes. The point of intersection is then the
solution of the system . Always check the apparent ordered pair solution in both equations. (See next
page.)

Example:

The plotting of these points may not be exact due to software limitations.

Check:

Thus X = -2 and Y = 3 is the solution to this system.

If the lines are parallel , there is no solution to the system. These systems are called inconsistent. If both
equations produce the same line, there are infinitely many solutions to the system. These systems are
called dependent.

Solving Systems of Equations by Elimination

A common algebraic method of solving systems of equations is the Elimination Method . In this method,
each equation is first multiplied by a non-zero number so that the sum of the coefficients on either X or
Y is zero. The equations are then added together producing a new equation in one variable . This equation
is solved, and the solution substituted back into one of the original equations to obtain the solution value
of the other variable.

Example:

Multiply the first equation by -3 and the second by 2 so that the sum of the coefficients on the X
variable will be zero.

Add equations Divide by 7

Substitute Y = 1 into the first equation and solve for X.

	First equation
	Substitute for Y and Multiply
	Add 5 to both sides
	Divide by 2

Check:

Thus the solution to this system is X = 3 and Y = 1

Solving Systems of Equations by Substitution

Another common algebraic method for solving systems of equations is the Substitution Method. In this
method one equation is solved for one of the variables. This algebraic expression is then substituted into
the other equation, producing a new equation in a single variable. This new equation is solved and the
resulting value substituted back into an original equation to solve for the other variable

Example:

Since the first equation is already in the form solving for Y, 3X – 7 will be substituted for the Y in
the second equation.

	Second equation
	Substitute 3X — 7 for Y
	Multiply & combine like terms
	Add 28 to both sides
	Divide by 9

Now that a value for X has been found, substitute this value back into the first original equation to
obtain a value for the Y variable.

	First equation
	Substitute 4 for X
	Multiply combine like terms

Check:

Thus the solution to this system is X = 4 and Y = 5.

Exercises

1. Solve by graphing	2. Solve by Addition Method	3. Solve by Substitution

4. A motorboat traveling with the current went 36 miles in two hours. Against the current it took an hour
longer to go the same distance. Find the rate of the boat in calm water and the rate of the current.

5. A coin bank contains only nickels and dimes. The total value of the coins is $2.50. If the nickels were
dimes and the dimes were nickels, the value of the coins would be $3.50. How many nickels are in the
bank?

6. A chemist has two alloys, one of which is 10% gold and 15% lead, the other of which is 30% gold and
40% lead. How many grams of each of these two alloys should be used to make an alloy that contains 60
grams of gold and 88 grams of lead?

7. How much water should be evaporated from 75 ounces of a 2% saline solution to produce a 5%
solution?

8. The difference between the ages of an oil painting and a watercolor is 35 years. The age of the oil
painting five years from now will be twice the age of the watercolor five years ago. Find the age of each
painting.

9. Two people, one rollerblader and one race walker, are 18 miles apart. They will meet in two hours if
they head toward each other. They will meet in four hours if they head in the same direction – that is – if
the rollerblader heads toward the walker. Find the rate of each.

ANSWERS

Equations and Inequalities

Exponents and Polynomials

Factoring Polynomials

Rational Expressions

Roots and Radicals

Graphing

The points should be plotted at (0, -2) & (3, 0).	The points should be plotted at (0, 3) & (-3, 0).

Systems of Equations