Download Strategy for Solving Algebraic Equations Example: 3(x

Strategy for Solving Algebraic Equations
Example:
3(x - 4) + 6 = 18 - 5x
1. Use the distributive property to remove parentheses:
3(x - 4) + 6 = 18 - 5x becomes 3x – 12 + 6 = 18 – 5x
2. Combine like terms on either side of the equation. -12 and 6 are like terms. -12 + 6 = -6
3x – 12 + 6 = 18 – 5x
3x - 6
= 18 – 5x
3. Use the addition or subtraction properties of equality to get the variables on one side of the = symbol and the
constant terms on the other.
3x and 5x are like terms. On the right side we have -5x so add 5x to both sides to get all the variable
terms on the left side.
5x + 3x – 6 = 18 – 5x + 5x
8x – 6 = 18
4. Continue to combine like terms whenever possible.
6 and 18 are like terms. Since 6 is subtracted from 8x, add 6 to both sides to move it to the other side.
8x - 6 + 6 = 18 + 6
8x = 24
5. Multiply both sides of the equation by the RECIPROCAL of the COEFFICIENT to isolate the variable.
The coefficient of x is 8. The reciprocal of 8 is ⅛
1
1
(8 x)  24 
8
8
x=3
6. Check the results by substituting your found value for x into the original equation.
3(x -2) + 5x = 18
3(3-2) + 5(3) = 18 ?
3(1) + 5(3) = 3 + 15 = 18 ? Yes. So x = 3 is the solution to the equation.
Translating Verbal Expressions into Mathematical Expressions
Verbal Expressions Examples
Math Translation
Addition
added to
more than
the sum of
increased by
the total of
6 added to y
8 more than x
the sum of x and z
t increased by 9
the total of 5 and y
6+y
8+x
x+z
t+9
5+y
Subtraction
minus
less than
subtracted from
decreased by
the difference
between
x minus 2
7 less than t
5 subtracted from 8
m decreased by 3
the difference between y and 4
x-2
t-7
8-5
m-3
y-4
10 times 2
one half of 6
the product of 4 and 3
10 X 2
(1/2) X 6
4X3
multiplied by
y multiplied by 11
11y
divided by
the quotient of
x divided by 12
the quotient of y and z
x/12
y/z
the ratio of
the ratio of t to 9
t/9
Power
the square of
the cube of
squared
the square of x
the cube of z
y squared
x
3
z
2
y
Equivalency
equals
is
is the same as
1+2 equals 3
2 is half of 4
½ is the same as 2/4
1+2 = 3
2 = (½)4
yields
represents
3+1 yields 4
y represents x+1
3+1 = 4
y=x+1
greater than
less than
greater than or equal
to
at least
no less than
less than or equal to
at most
no more than
-3 is greater than -5
-5 is less then -3
x is greater than or equal to 5
-3 > -5
-5 < -3
x≥5
x is at least 80
x is no less than 70
x is less then or equal to -6
y is at most 23
y is no more than 21
x ≥ 80
x ≥ 70
x ≤ -6
y ≤ 23
y ≤ 21
Multiplication times
of
the product of
Division
2
1
2

2
4
Comparison
Consecutive consecutive integers the sum of two consecutive integers is 3
consecutive odd
integers
the product of two consecutive even
integers is 8
n + (n +1) = 3
n(n+2) = 8
Solving Application Problems
Problem-Solving Strategy:
•
Analyze the problem. What are you trying to find? What’s the given info? Label variables to the
unknown quantities. Use algebraic expressions to put one unknown in terms of the other, if possible.
•
Work out a plan before starting. Draw a sketch if possible. Look for indicator words (e.g. gained, lost,
times, per) to know which operations (+,-, x,÷) to use.
•
Solve the problem.
•
Check your work. If the answer is not reasonable, start over.
Example 1:
The difference of two numbers is 3. Their sum is 13. Find the numbers
1) What are we trying to find? Two numbers whose difference is 3 and whose sum is 13.
2) Given info: Difference = 3, Sum =13. Let variables represent the unknown numbers. x = 1 st number, y = 2nd number.
3) Work out a plan. Use the variables in equations using the given info. The “Difference” means subtraction, x – y = 3.
“Sum” means addition, so x + y = 13
4) Solve the equations
x–y=3
x + y = 13
You can solve by elimination if you add the two equations together.
2x+ 0 = 16
2x = 16
x=8
Don’t forget to solve for y! x + y = 13…… 8 + y = 13 …… y = 5
5) Check your work.
Does 8 – 5 = 3? Yes
Does 8 + 5 = 13? Yes
Answer: The two numbers are 8 and 5.
Example 2:
Find three consecutive even integers whose sum is 72.
1) What are we trying to find? 3 consecutive even integers whose sum is 72
2) Given info: Sum = 72. Each integer is 2 units apart.
3) Work out a plan. Let variables represent the unknown numbers.
n = 1st number, n + 2 = 2nd number, (n+2) + 2 = 3rd number, which can be simplified to n + 4.
Use these variable expressions in equations using the given info. “Sum” means addition, so
n +( n+2) + (n+4) = 72
4) Solve the equation
3n + 6 = 72
3n = 66
n = 22
Don’t forget to solve for all three numbers! n = 22, n + 2 = 24, n+4 = 26
Answer: The three integers are 22,24, and 26
5) Check: Does 22 + 24 + 26 = 72? Yes
Example 3:
On a night when they scored 110 points, a basketball team made only 5 free throws (worth 1 point each). The
remainder of their points came from two- and three-point baskets. If the number of baskets from the field
totaled 45, how many two-point and how many three-point baskets did they make?
What are we being asked to find? The number of two- and three-point baskets.
Let x = # of three-point baskets
Let y = # of two-point baskets
Given Info:
They scored 110 points.
5 of those points were from free throws.
The number of baskets from the field was 45.
Form equations:
The number of baskets = # of three point baskets + # of two-point baskets.
x + y = 45
Putting y in terms of x gives:
y = 45 - x
Points from three point baskets = (3 pts per basket)(x baskets) = 3x
Points from two-point baskets = (2 pts per basket)(45-x baskets) = 2(45-x)
Points from free throws = 5 (this was given)
Equation for Total Points: 110 = 2(45 – x) + 3x + 5
110 = 90 – 2x + 3x + 5
110 = 95 – 2x + 3x ….Now think of this as 95 + -2x + 3x
110 = 95 + x
110 = 95 + x
-95
(Subtract 95 from both sides)
-95
15 = x (remember x represents # of 3-pt baskets and y = 45-x = 45-3 = 42 = # of 2-pt baskets)
Check: Does 110 = 2(45 – x) + 3x + 5?
110 = 2(45 – 15) + 3(15) + 5 = 2(30) + 45 + 5
110 = 110 Yes
Conclusion:
The team made 15 three-pt baskets and 42 two-pt baskets.