Linear equations can form a system if two or more equations contain the same linear variables. A solution to a system of linear equations must be correct for both equations in the system. If a value solves one equation in the system but not the other, it is not a solution.

A particular ordered pair can be tested within each equation in a linear system to see if it is a solution to the equations in the system. Suppose the ordered pair is (1, 3) and the equations in the system are 2x +y = 5 and -2x +y =1. In the first equation, 2(1) is 2 and 2 +3 = 5, so (1, 3) is a solution of the first equation. In the second equation, -2(1) is -2 +3 is 1, so (1, 3) is a solution of the second equation, thus a solution of the system. Suppose the ordered pair is (2, -1) and the equations are x -2y = 4 and 3x +y = 6. In the first equation 2 -2(-1) = 2 +2 =4, so the ordered pair (2, -1) is a solution of the first equation. In the second equation 3(2) is 6 -1 is equal to 5, not to 6. The ordered pair (2, -1) is not a solution of the system, because it doesn’t solve both systems.

If a system of linear equations has one solution, the graph of each linear equation in the system will intersect at only one point. Suppose the equations are 2x +2y =6 and 4x-6y =12. They appear from the graph to have one solution, the ordered pair (3, 0). However, the only way to see if that solution is correct is to substitute the ordered pair in each equation to determine if it is a solution of the system. In the first equation, 2(3) + 2(0) =6, so the ordered pair is a solution. In the second equation, 4(3) -6(0) is 12, so that ordered pair (3, 0) is a solution of the system.

Sometimes a system of linear equations has more than one correct solution. The only time a linear system can have all solutions in common is if each linear equation is the same line. The solutions are infinitely many, because any solution of one equation is also a solution of the other equation. For example, a system of equations such as 2x –y =3 and 4x – 2y = 6 both cover the same line.

Some systems of linear equations have no solutions in common. If two lines have the same slope but different y intercepts, they will graph as parallel lines. They can be graphed as a system of linear equations, because they have the same variables, but any solution for one linear equation will not be a solution for the other linear equation.

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]]>The graphs of linear functions can be transformed without changing the shape of the line by changing the location of the y intercept or the slope of the line. Those lines can be transformed by translation, rotation, or reflection, and still follow the slope-intercept form y = mx + b.

The graphs of linear functions all have the same basic shape: a line. They form a family because they all follow that basic shape and their lines express the same equation, y = mx + b, the slope-intercept form. The most basic function in the family, f(x) =x, is called the parent function.

One line can be translated to another line if the y-intercept is changed. Since the slopes of the lines are still the same, but they cross the y axis at different points, they are parallel. Suppose one line has the equation y = x + 3 and another line has the equation y = x – 2. The slope of the line x is the same for both lines, but one crosses the y axis at 3, while the other crosses the y axis at -2. It is as if the entire line with the equation y = x + 3 slid down to fit in the position y = x – 2. The line y = x is parallel to the other two, as the slope x is still the same, but the y intercept is at the origin.

Graphs of lines can also be changed so they have the same y intercept, but different slopes. The line may have a steeper slope or one that is less steep. It is as if the entire line were turned. For example, lines of direct variation have the same y-intercepts but different slopes. All the lines may have their y intercepts at the origin, but the line y = -3x is steeper than the line y = 2x, and the line y = x is not quite as steep.

Remember that the reflection of a figure is its opposite across a line of symmetry. The reflection of a line is similar to the reflection of a figure, as it represents its mirror image. If a line has slope m, its reflection will have a slope multiplied by -1. Suppose a line can be represented by the equation y = 3x. Its reflection will have the slope (3) (-1), or -3.

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]]>Writers use different points of view in essays or stories in order to present and reveal information to their readers. They may choose first-person, second-person, or two types of third-person narrative devices.

First-person narratives tell the story from inside the mind and feelings of a chosen narrator, using pronouns such as I, me, mine, we, and us. For example, “During my summer vacation, I went up in a hot-air balloon. It was my first flight, and I was so excited to watch the ground grow smaller and further away.”

Second person narratives are directive, using pronouns such as you and yours. The reader is drawn into the action and told to do things. It is rarely used in essays or stories, but is usually limited to instructional material and directions. “Imagine that you went on a hot-air balloon ride during your summer vacation, and it was your first flight. What would you see, feel, hear, taste and smell during the trip? Describe your feelings and emotions in an essay.” When you write that essay, you will not answer the questions in the second person.

Third-person narratives tell the story in a more detached way, using pronouns such as he, she, him, her, it, they, and them. When the point of view is limited, the perspective is from one character’s point of view, limited to what that character is thinking, feeling, and observing. That character may be the main character (protagonist) or one of the secondary characters, as long as that character can observe the actions of others. For example, “Jill spent her summer vacation in Albuquerque, New Mexico, and rode on a hot-air balloon for the first time. She was excited about the opportunity, and often reflected on it as the high point of that summer. She watched over the side of the basket as the people waiting on the ground grew smaller and smaller.”

Third-person omniscient point of view uses the same pronouns, but differs in perspective. The narrator is not limited to one characters’ point of view, but can also tell what other characters are thinking and feeling. In the third-person omniscient point of view, the narrator can tell how Jill feels on her first hot-air balloon flight, talk about what Jane is thinking as she waits on the ground for her turn, and what the more experienced passengers in the balloon are thinking and feeling.

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]]>Lines representing graphs of linear equations have slopes, defined as the change in y/change in x. Parallel lines have the same slope, but no solutions in common. Perpendicular lines intersect at one point to form right angles.

Imagine a line graphing the equation y = 2x +3. Using the slope-intercept formula y = mx +b, the equation has a slope of 2. Suppose another line exists graphing the equation y = 2x +5. Using the slope –intercept formula y = mx +b, that equation also has a slope of 2. The lines have no points in common, and they both have the same slope. Those lines are parallel. Lines that are not vertical are parallel if they have the same slope. Vertical lines, which have an undefined slope, are parallel by definition, as they do not intersect.

Lines that are perpendicular intersect at one point to form right angles, measuring 90 degrees. Not all lines that intersect are perpendicular. Vertical lines are perpendicular to horizontal lines by definition. For example, lines represented by the equations y = -2 and x = 4 are perpendicular. Lines that are not vertical are perpendicular only if the product of their slopes equals -1. Suppose a line y = 2/3x +1 and another line y= -3/2x +2. They will meet at one point and be perpendicular to each other. The product of their slopes (2/3) (-3/2) equals -1.

Sometimes math problems present the coordinates of points along a line, rather than the slope-intercept formula. The slopes of the lines can be calculated using the slope formula, then compared. If the slopes are the same, they are parallel. If the product of their slopes equals -1, they are perpendicular. Suppose one line passes through the points (0, 2) and (-3, -3) and another line passes through the points (4, 2) and (1, -3). The slope of the first line is given by the equation for the slope (-3 -2)/ (-3-0) or -5/-3. The slope of the second line is given by the equation (-3-2)/(1-4) or -5/-3. The lines are parallel.

The algebra of slopes of parallel and perpendicular lines can be applied to geometric forms. Since parallel lines have the same slope, lines of a parallelogram can be shown to be parallel. Suppose the coordinates of point A are (0, 2); point B (4, 2); point C (1, -3), and point D (-3, -3). A figure drawn on ABCD is a parallelogram, because lines AD and BC are parallel, and lines AB and CD are parallel.

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]]>A direct variation is a linear relationship between variables so they have a constant ratio. It is a special case of the slope-intercept form y =mx +b, where b = 0.

Suppose that Papa’s Pizzeria has a lunch special on pizza, so that one slice sells for $2.00, 2 slices sell for $4.00, 3 slices sell for $6.00 and so on. If the number of slices is represented by x, the cost in dollars is represented by y. The relationship between x and y can be represented by the equation y= 2x. It is a direct variation in the form y=kx, where k is the constant of variation. In this case, the constant of variation equals 2.

One of the ways to identify direct variation is to look at the equation and determine if it follows the form. An equation such as y=4x follows the form y=kx, with the constant of variation equaling 4. In order to solve an equation such as -3x +5y =0 for y, first add 3x to both sides, such that -3x +3x +5y = 0 +3x. That leaves the equation 5y=3x. Then both sides can be divided by 5, so that (5y)/5 = (3x)/5. That leaves an equation such as y=3/5x. It is also in the form y=kx, and in this case, the constant of variation equals 3/5. However, an equation such as 2x +y = 10 does not follow the form.

Another way to look at direct variation is to determine if the ratio y/x is a constant for all values of the variable. Suppose that the ordered pairs include the set {(1, 5) (2, 10) (3, 15) (4, 20)}. The ratio of y/x in the first ordered pair is 5/1. The second pair, 10/2, reduces to 5/1. The third pair, 15/3, also reduces to 5/1, as does the fourth pair, 20/4.

Any linear equation can be expressed in the form y = mx +b, where m is the slope and b is the intercept. Recall that the slope of a line is the measure of the rate of change, and that any point on the line in a linear relationship will have the same slope. Suppose that the equation is y=2x +1. Two of the ordered pairs on the line have the values (2, 5) and (1, 3). When the line is graphed, the slope is the same at any point on the line.

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]]>Surface processes such as weathering and erosion transform mountains and other landforms. Rocks and minerals disintegrate into sediment, and it is transported into other areas, building up new landforms.

At the surface, rocks and minerals crumble into smaller pieces from a variety of forces. Many of these forces, known as mechanical weathering, do not change their chemical composition. For example, when water freezes in a crack in a boulder, ice expands to widen the crack. When the ice thaws, the boulder fractures into smaller rocks. Rocks can also be ground into smaller pieces by friction when other rocks rub against them, driven by wind or water.

Chemical weathering changes the chemical composition of the rocks and minerals themselves. Corrosive chemicals in water as a result of pollution can eat away at rock. Although natural rainfall is slightly acidic, sulfur dioxide and nitrogen oxides combine with water as acid rain. Oxygen in the air rusts metals. Feldspar in granite turns to clay when it combines with water. Clay takes up more space as it expands, resulting in mechanical weathering of the remaining rock, similar to that caused by ice.

Weathered rock is carried from place to place by erosion. For example, glaciers carry rocks down mountain slopes as they flow and melt. Rocks are ground into smaller pieces, and the sediment is transported as debris. Water in rivers and streams bring sediment long distances, depositing it to form river deltas. Streams and rivers carve canyons through cliffs of sedimentary rock over time. Wind blows sand across rock, sandblasting patterns into cliff faces in desert areas.

The greatest destruction from erosion occurs when steep slopes give way, called mass wasting. Most of the time, rock cliffs or soil are fairly stable, held in place by gravity. If the area is undermined by flowing streams or saturated by heavy rainfall, however; sudden landslides careen down unstable slopes. Avalanches of rocks, snow, and other debris devastate the steepest slopes regularly. Earthquakes and volcanic eruptions can trigger huge landslides, as the tilted faults become unstable.

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]]>Mountains form as a result of intense tectonic forces. Mountain chains such as the Andes and the Himalayas rise from the collisions of continental plates, as rocks are folded, uplifted, and faulted.

Mountains have peaks that are at higher elevation than the surrounding land. The definition of a mountain usually includes the height of the peak as well as the slope of the elevation leading up to the peak. The lowest mountains may only be 300 meters higher than the surrounding land but have a steep elevation to the peak. A mountain with a peak at 1000 meters has a slope greater than 5 degrees, a mountain with a peak of 1500 meters has a slope greater than 2 degrees, and a mountain with a peak of 2500 meters will have a slope greater than 1 degree. By comparison, Mount Everest is 8,848 meters above sea level.

Tectonic plates collide to form most mountain chains. Trapped magma forced to the surface creates volcanic ridges and chains such as the Pacific “Ring of Fire.” Rock layers can be folded by intense pressure from either side. Faults raise and lower blocks of the earth’s crust, sometimes violently in earthquakes. Those blocks can be tilted and reversed as they are further deformed.

The Andes Mountains, in western South America, were formed as a result of collisions between oceanic plates and continental plates a hundred million years ago. Magma erupted to create volcanoes as the oceanic plate dove under the continental plate. Some of the igneous rock was further transformed by faulting and folding into sedimentary and metamorphic rock.

The Himalaya Mountains, between India, Nepal, Bhutan, Tibet, China, and Pakistan, were formed as a result of collisions between continental plates, roughly 75 million years ago. Scientists believe that the Eurasian and Indian plates collided. However, the rocks forming continental plates were at the same density, so one could not dive under the other. They faulted and folded instead, creating a mountain range with some of the tallest mountains in the world, including Mount Everest and K2. The entire region is at a very high elevation.

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]]>Writers can make essays more vivid by providing specific, concrete examples for abstract concepts and language. Effective writing requires a balance between the two. Abstract concepts without development are unclear, but a list of concrete examples appear disorganized.

Abstract language includes words that express thoughts and ideas, such as bravery, freedom, love, and compassion. Thoughts and ideas are intangible, but they are essential to any type of connected essay. Without thoughts and ideas, the essay will lack structure. Suppose the writer is presenting an argument that arts education enhances development. The idea of arts education is fairly broad and abstract, as is the concept of development. Unless those concepts are defined further, readers will not be able to follow the argument or draw their own conclusions.

Concrete language, by contrast, includes words that can be touched, seen, heard, felt, or even smelled. The wildfire scorched thousands of acres. The brave firefighter rescued a woman who was trapped inside the burning house. The acrid smoke stings the eyes. The crackling flames rose higher as the fire raged out of control. The blackened hillside showed evidence of the fire’s destruction. While bravery is an abstract term, the concrete example describes a brave action. Similarly, the descriptions of the wildfire present concrete details to make it more vivid.

General language requires clarification. A writer who begins a review with “The book is boring” must give specific details to support that opinion. Perhaps the book is boring because the plot moves slowly and the characters are stereotyped. Similarly, a statement like “That film was interesting” doesn’t give enough reasons for someone else to want to see it. A sentence such as “The boat floated down the river” doesn’t present any details as to which boat floated down the river.

Specific language is vivid and clear. The sailboat with the red sail floated down the Columbia River when it came untied from its mooring. The piano concert begins at Benaroya Hall at 7:00 tonight.

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]]>Wordiness errors obscure the writer’s message under unnecessary verbiage, such as redundant words or phrases, passive voice, weak “to be” verb forms, and unnecessary expressions. Close reading and editing can eliminate many of them before an essay is in final form.

Some types of wordiness include redundant words or phrases. In the sentence, “The balloons were red in color,” red is already a color. Simply state that the balloons were red. Similarly, the phrase “snow continued to keep on falling” could be more simply stated as “snow continued falling.” Something continued is also keeping on. “We left the park because of the rainy weather” could be recast as “We left the park because of the rain.”

Compare a sentence such as “The boat was sailed by John” with “John sailed the boat.” The first sentence is in the passive voice, while the second sentence is in the active voice. In the passive voice, the one doing the action is obscured into a prepositional phrase. Both sentences describe the same activity, but the sentence in the active voice expresses direct action with fewer words.

Compare sentences such as “The cost of the book is twenty dollars” with “The book costs twenty dollars.” Both sentences describe the same activity, but the first sentence uses a static form of the verb “to be”, while the second sentence replaces **is** with **costs**. Sentences using passive construction add a form of the verb “to be” as an auxiliary verb. During a final edit, check sentences for forms of the verb “to be” to make sure that they are not in the passive voice or hiding the action.

Some sentences have phrases that add nothing to the meaning of the sentence. A sentence such as “it is a fact that some phrases can be excised from a sentence without changing its meaning” can be recast to “some phrases can be excised from a sentence without changing its meaning.” Other unnecessary phrases include “the reason why is that,” “in the truest sense of the word,” “in the event that,” “for the purpose of,” and “in the event that.”

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]]>The rate of change is a ratio that compares the change in values of the y variables to the change in values of the x variables. If the rate of change is constant and linear, the rate of change is the slope of the line. The slope of a line may be positive, negative, zero, or undefined.

Another way to look at the rate of change is to look at the definitions of the variables themselves. The y variables are the dependent variables and the x variables are the independent variables. Suppose that a worker is paid $10.00 per hour. The amount of the paycheck depends on the number of hours worked, so that worker is paid $160.00 for 16 hours one week, and $180.00 for 18 hours the next week. The change in the dependent variable is 180-160 or 20 and the change in the independent variable is 18-16, or 2, so the rate of change is 20/2, or 10/1.

If the relationship between the dependent and independent variables is linear, the rate of change is the same between any two sets of variables along the line. Suppose that the worker in the above example is paid $50.00 for 5 hours one week and $80.00 for 8 hours the next week. The change in the dependent variable is 80-50, or 30, and the change in the independent variable is 8-5, or 3. The rate of change is 30/3, or 10/1, the same rate of change as in the first example. In a linear relationship, the rate of change is constant.

The slope of a line can also be described by its direction. In the relationship of the paycheck and number of hours worked, the line rises from left to right. The more hours worked, the higher the paycheck, with a positive slope. Suppose a plane is landing. The relationship between its elevation and the time from its highest altitude is a falling line from left to right, a negative slope. If the relationship is a horizontal line, so that no change occurs, the slope is zero. However, if the relationship is a vertical line, the slope is undefined. It is also not a function, as there are multiple values of y for one value of x.

The slope of a line is usually represented by the variable m. It is expressed by the ratio of the difference in value of y variables to the difference in value of x variables. In algebraic terms, if (x_{1}, y_{1}) (x_{2}, y_{2}) are the coordinates of any two points on a line, then m= (y_{2} – y_{1})/(x_{2} – x_{1}).

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