LESSON 1-4A: SOLVING EQUATIONS
[TEXT: LESSON 1-4]
Review: Solve each equation for x. Show all work/steps vertically.
1) 7x – 5 = 4x + 10
2) 2(3x – 4) + 3x = 2(x + 7) – 4
3) 11 + 3x – 7 = 6x + 5 – 3x
4) 6x + 5 – 2x = 4 + 4x + 1
Practice: Is the equation always, sometimes or never true? How many solutions are there to each
equation? What are the solution(s) to each equation?
5) 2x + 3(x – 4) = 2(2x – 6) + x
6) 7x + 6 – 4x = 12 + 3x – 8
An equation may have one, none, or many (infinite) solutions.
* An equation has no solution if no value of the variable makes the equation true.
* An equation that is true for every value of the variable is an identity and has infinitely many
solutions.
SOLVING EQUATIONS WITH FRACTIONS
Tip: Simplify the equation by multiplying each and every term of the equation by the common
denominator of all fractions within the original equation.
Examples: Solve each equation for x.
EX 1:
–3= +6
EX 2:
–2= +
Practice: Solve each equation for x.
Tip: Simplify the equation by multiplying each and every term of the equation by the common
denominator of all fractions within the original equation.
7)
+5= –2
8)
+2= +
9)
–
=2
LESSON 1-4B: SOLVING LITERAL EQUATIONS
A literal equation is an equation that uses at least two different letters as variables. You can solve a
literal equation for any one of the variables "in terms of" the other variable(s). Solutions to literal
equations will not be a constant because they will include the other variables in the equation. Formulas
with at least two variables are examples of literal equations.
Examples of Literal Equations: 3a + 5b = 7c,
Formulas:
P = 2(w + l),
, 2x + 3y = 12,
F = ma, E = mc2, V =
, A = bh
Solve literal equations using the same process for solving a basic equation in one variable. Your
solution(s) will contain the other variables in the equation.
Examples: Solve each equation for the specified variable. Show work vertically.
EX 1: 2x + 3y = 12 for y
EX 2:
A=
for b
EX 3: V = bh for h
EX 4: h = –10t2 + vt for v
EX 5: The equation C = (F – 32) relates temperatures in degrees Fahrenheit F and degrees Celsius C.
What is F in terms of C?
Practice: Solve each equation for the specified variable. Show work vertically.
1) 4x – 3y = 15 for y
2) P = 2(l + w) for w
3) V = hr3 for h
4) A = r2
for r
SOLVING EQUATIONS WITH THE VARIABLE IN MULTIPLE TERMS
Tip: Re-organize equation so that the variable in which you are solving is on the same side of the
equation. Factor out the common factor (variable in which you are solving) and then divide by the
other factor to get your solution(s).
Examples: Solve each equation for the specified variable. Show work vertically.
EX 6: Solve for x:
ax + bx = c – d
EX 7:
Solve for a:
ax – by = 2a + 7
Practice: Solve each equation for the specified variable. Show work vertically.
5) Solve for x:
2a + 3x = bx – 1
7) Solve for x: ax + by = 3c – 5x + 7
6)
Solve for a: 5c + ab = 2a + 4d
8) Solve for a:
–5=
+2
ASSIGNMENT
LESSON 1-4A: SOLVE EQUATIONS
1-6: Solve each equation. Show work vertically. Do not round answers. Check solutions.
1) 7x + 4 = 3 – 2x
4)
+3= –1
2) 6(n – 4) = 3n
3) 4x + 2(x + 5) = 8x – 14
5)
6)
+4= +
–
=2
7-8: Identify the error – Circle the step that contains the error in each of the following. Rework the
problem correctly, showing all steps of work.
7)
+1= +
8)
Solution/work:
6
12
–
=4
Solution/work:
12
4
12
+ 1 =
3
12
3
15
5
15
–
+
15
= 4
3x – 5x –10 = 60
3x + 12 = 4x + 3x + 1
–2x – 10 = 60
3x + 12 = 7x + 1
–2x = 70
11 = 4x
x = –35
=
=x
x=
or 2 or 2.75
9-11: Is the equation always, sometimes or never true? How many solutions are there to each
equation? What are the solution(s) to each equation?
9) 4x + 3(x – 2) = 2x + 5(x + 1)
10) 4(3x + 5) = 2(6x + 10)
11) 3(x + 2) + 5 = 4x – 7
ASSIGNMENT
LESSON 1-4B: SOLVE LITERAL EQUATIONS
1-15: Solve each equation for the specified variable. Show work vertically.
1) 3x – 2y = 8 for y
2) –5x + 3y = 15
3) 3a + 4b = 5c – 6d for a
4) A = bh for h
5) s = gt2 for g
6) V = lwh for w
7) C = 2r for r
8) h = –5t2 + vt for v
9) h = –5t2 + vt for t
11) A = h(
12) ax – 3b = 4c + 7x for x
10) A = h(
+
) for h
13) 2(a + 3b) = ax – 4b for a
14)
– 4b =
+
) for
+2
15)
= +
for a
16: The solution to the following equation is given. Show the correct and complete work to get to
this solution.
– 3c =
Equation:
Solution:
x=
for x