Solve The Equation Below. Show All Work Following The Method

Solve the Equation Below Show All work following the methods discussed in class

The assignment requires solving various algebraic equations, inequalities, and word problems, including linear equations, inequalities, graphing lines, and application problems involving growth and proportions. The solutions should show all steps with proper justification and include checks for equations with unique solutions. Additionally, some problems involve translating between different forms of expressions, such as scientific notation, and performing operations like simplifying algebraic expressions and working with percents and fractions.

Paper For Above instruction

Algebraic problem-solving is fundamental in understanding mathematical relationships and practical applications, such as budgeting, growth modeling, and scientific measurements. In this comprehensive analysis, we will explore methods to solve linear equations and inequalities, analyze word problems that translate to algebraic expressions, and interpret data through graphing and modeling.

Solve the given linear equations and verify solutions

The first problem involves solving the linear equation 7(4x + 5) = -5 + 8(8x). Expanding both sides gives:

28x + 35 = -5 + 64x

Bring variables to one side:

28x - 64x = -5 - 35

-36x = -40

Solve for x:

x = (-40) / (-36) = 10 / 9 ≈ 1.111...

To verify, substitute x = 10/9 into the original equation:

Left side: 7(4(10/9) + 5) = 7(40/9 + 5) = 7(40/9 + 45/9) = 7(85/9) = (785)/9 = 595/9

Right side: -5 + 8(8(10/9)) = -5 + 8(80/9) = -5 + (640/9) = -45/9 + 640/9 = 595/9

Both sides equal 595/9, confirming the solution.

Next, solving 48 - 5(1 - n) = 3(4 - 2n) - 2:

Expand both sides:

48 - 5 + 5n = 12 - 6n - 2

Combine like terms:

43 + 5n = 10 - 6n

Bring variables to one side:

5n + 6n = 10 - 43

11n = -33

n = -3

Check by substituting n = -3 back into the original:

Left: 48 - 5(1 - (-3)) = 48 - 5(4) = 48 - 20 = 28

Right: 3(4 - 2(-3)) - 2 = 3(4 + 6) - 2 = 3(10) - 2 = 30 - 2 = 28

Equality holds, so the solution is valid.

Solving for y in linear equations

The equation 5(y + 4) - 2(1 - 3y) = 11y - 7 expands to:

5y + 20 - 2 + 6y = 11y - 7

Combine like terms:

(5y + 6y) + (20 - 2) = 11y - 7 → 11y + 18 = 11y - 7

Subtract 11y from both sides:

18 = -7

This is a contradiction, indicating the original equation has no solution.

Solve for m in the equation 17m + 1 = 2m + ?

Since the right side appears incomplete in the prompt, assuming the full equation is 17m + 1 = 2m + c, where c is a constant, then solving for m:

17m - 2m = c - 1

15m = c - 1

m = (c - 1) / 15

Without a specific constant, the solution is expressed in terms of c.

Word problem involving Congress members

Let D be the number of Democrats, R be the number of Republicans:

R + D = 435

R = D + 57

Substitute into the first equation:

(D + 57) + D = 435

2D + 57 = 435

2D = 378

D = 189

Then R = 189 + 57 = 246

There are 189 Democrats and 246 Republicans.

Travel time problem involving Janet and Bob

Janet departs at 6:30 am traveling at 24 mph, so her distance after t hours is:

d_J = 24t

Bob departs 2 hours later, at 8:30 am, traveling at 40 mph. The time Bob travels is (t - 2) hours. His distance is:

d_B = 40(t - 2)

Bob catches up when d_J = d_B:

24t = 40(t - 2)

24t = 40t - 80

80 = 40t - 24t = 16t

t = 80 / 16 = 5 hours

Janet started at 6:30 am, so Bob catches her at:

6:30 am + 5 hours = 11:30 am.

Solve inequalities and graph solutions

For the inequality 9x - 3 ≤ 17x - ? (assuming a missing constant), let's solve 9x - 3 ≤ 17x:

Subtract 9x from both sides:

-3 ≤ 8x

x ≥ -3/8

Expressed in interval notation: [ -3/8, ∞ ).

Graphically, on a number line, the solution includes all numbers from -3/8 to infinity, including -3/8.

For the compound inequality −23 ≤ 7x - 2

Add 2 to all parts:

-21 ≤ 7x

Divide all parts by 7:

−3 ≤ x

Solution in interval notation: [−3, 2)

This denotes all real numbers from -3 (inclusive) up to but not including 2.

Graphing linear equations and lines

Given the linear equation 3x + 2y = 0, find three solutions:

Let x = 0: 2y = 0 → y = 0 → (0, 0)

Let x = 1: 3(1) + 2y = 0 → 3 + 2y = 0 → 2y = -3 → y = -3/2 → (1, -3/2)

Let x = -1: 3(-1) + 2y = 0 → -3 + 2y = 0 → 2y = 3 → y = 3/2 → (-1, 3/2)

Plot these points to graph the line.

Line parallel to y-axis through point (4, 5)

The line is vertical at x = 4. Its equation is x = 4, and the slope is undefined.

Slope and intercept of the line 7x - 5y = -15

Rewrite in slope-intercept form:

-5y = -7x - 15

y = (7/5)x + 3

Slope: 7/5

Y-intercept: (0, 3)

Line through points (-3, 5) and (6, -2)

a) Slope:

m = (−2 − 5) / (6 − (-3)) = (−7) / 9 = -7/9

b) Equation in point-slope form (using point (-3, 5)):

y - 5 = (-7/9)(x + 3)

c) Convert to slope-intercept form:

y = (-7/9)(x + 3) + 5 = (-7/9)x - 7/3 + 5 = (-7/9)x + (15/9 - 7/3)

Express 5 as 15/3:

y = (-7/9)x + (15/3 - 7/3) = (-7/9)x + (8/3)

d) Convert to standard form: multiply through by 9 to clear denominators:

9y = -7x + 24

Rearranged:

7x + 9y = 24

e) Graph the line using the intercepts or the points derived above.

Perpendicular line through (-5, 2) to 4x + 3y = 7

The slope of the original line:

3y = -4x + 7 → y = (-4/3)x + 7/3

The slope of the perpendicular line is the negative reciprocal:

m = 3/4

Line through (-5, 2):

Using point-slope form:

y - 2 = (3/4)(x + 5)

or in slope-intercept form:

y = (3/4)(x + 5) + 2

This line is in point-slope form with slope 3/4 and passes through the given point.

Modeling Starbucks store growth with a linear equation

Points (0, 8896) and (9, 11965) describe the number of stores over years. Find the slope:

m = (11965 - 8896) / (9 - 0) = 3069 / 9 = 341

Line equation in slope-intercept form:

y = 341x + 8896

Predict stores in 2020 (x = 15, since 2005 is x=0):

y = 341(15) + 8896 = 5115 + 8896 = 14011

There will be approximately 14,011 Starbucks stores in 2020.

Number formatting and simplification

Express 2,354,107 in scientific notation:

2.354107 × 10^6

Express 0.000045 in scientific notation:

4.5 × 10^−5

Multiply (3x^7 y^8)(5x^9 y^12):

15x^{7+9} y^{8+12} = 15x^{16} y^{20}

Simplify (2x^5 y^{−3})^4:

2^4 x^{5×4} y^{−3×4} = 16 x^{20} y^{−12} = 16 x^{20} / y^{12}

Operations with algebraic expressions

Subtracting (4x^2 + 3x - 7) from (9x^2 + 2x + 3):

(9x^2 + 2x + 3) - (4x^2 + 3x - 7) = 5x^2 - x + 10

Multiplying (5x - 3)(8x^2 - 4x + 15):

40x^3 - 20x^2 + 75x - 24x^2 + 12x - 45 = 40x^3 - 44x^2 + 87x - 45

Additional calculations

Simplify 7x − 2 ≠ 12: the inequality 7x - 2 ≤ 12 implies:

7x ≤ 14 → x ≤ 2

Solution set: (−∞, 2]

Graph on the number line from minus infinity up to and including 2.

Conclusion

This detailed problem set demonstrates the application of algebraic techniques to solve equations, inequalities, and real-world problems. Each solution is verified through substitution, and solutions are expressed in appropriate formats, including interval notation and graphing. These methods are essential tools in mathematics and its applications, with broad relevance across disciplines such as science, economics, and engineering.

References

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