Praxis-II - Mathematics (5003)
Algebraic Thinking
An algebraic expression is a mathematical phrase that contains terms that include numbers (constants), variables, or a combination of numbers and variables.
A term in an expression is separated by a + or - sign.
Solve a system of equations by substitution:
- Solve one of the equations for either variable.
- Substitute the expression from Step 1 into the other equation.
- Solve the resulting equation.
- Substitute the solution in Step 3 into one of the original equations to find the other variable.
- Write the solution as an ordered pair.
The substitution method is the algebraic method to solve simultaneous linear equations. As the word says, in this method, the value of one variable from one equation is substituted in the other equation.
What is the substitution method?
Ans: The substitution method is a technique for solving systems of linear equations. Let's talk through a couple of examples.
Example:
I will be asked to solve this system of equations:
3x+y = − 3
x = −y + 3
The second equation is solved for x, so we can substitute the expression
−y+3 in for x in the first equation:
3x+y = -3
3(−y+3)+y = -3
−3y+9+y = -3
−2y = - 12
y = - 6
Plugging this value back into one of our original equations, say
x=−y+3x, we solve for the other variable:
x =−y+3
x =−(6)+3
x =−3
The solution to the system of equations is
x=−3x,
y=6y
We can check our work by plugging these numbers back into the original equations. Let's try
3x+y=−3
3(−3)+6 = -3
−9+6 = -3
−3
Want to learn more about the substitution method? Check out this video.
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An algebraic expression is just a collection of numbers and variables.
3x + 4y is an algebraic expression meaning
"3 times a number x plus 4 times another number y."
2x³ is an algebraic expression meaning" 2 times x times x."
It is important to use the correct names for different algebraic expressions.
Definitions:
Term:
An algebraic expression with numbers and variables connected only by the operations of multiplication and division.
3x, 2xy, - 7a² 4abc, 3/a, a²b/c²
Since terms are formed using only multiplication and division, plus and minus signs separate terms and make expressions with more than one term.
Monomial:
An algebraic expression with one, and only one, term.
2abfg, x²y², (2)(3)xyz, xyz/3
Binomial:
An algebraic expression with two, and only two, terms separated by either a plus sign or a minus sign.
2x - 3y, x² - 4y²
ab + 4a²
abcd + 5d
Trinomial:
An algebraic expression with three, and only three, terms, separated by either plus or minus signs.
x² + 2x - 7
3x - 4y + z
Polynomial:
Any algebraic expression with more than one term. Polynomial is a general name for binomials as well as trinomials, but not for monomials.
2x - 3y
abc - 7cd - 8ce
2x - 3y -4z + 4a - 2b
The special product of the sum of two numbers and the difference of the same two numbers equals the difference of the squares of these numbers.
(a + b) (a - b) = a² - b²
To the factor, in the difference between two square numbers, take the square root of the first number and the square root of the other number, and separate them, once with a plus sign and once with a minus sign.
a² - b² = (a + b)(a-b)
