Heart of Algebra
Example 4: To edit a manuscript, Miguel charges $50 for the first 2 hours and $20 per hour after the first 2 hours. Which of the following expresses the amount in dollars, C, Miguel charges if it takes him x hours to edit a manuscript, where x > 2?
Miguel charges pays for his first 2 hours of editing, he charges $20 per hour only after the first 2 hours. Thus, if it takes x hours for Miguel to edit a man uscript, he charges $50 for the first 2 hours and $20 per hour for the remain ing time, which is x − 2 hours. Thus, his total charge, C, can be written as C = 50 + 20(x − 2). This does not match any of the choices. But when the right-hand side of C = 50 + 20(x − 2) is expanded, you get C = 50 + 20x − 40, or C = 20x + 10
Example 7: Each morning, John jogs at 6 miles per hour and rides a bike at 12 miles per hour. His goal is to jog and ride his bike a total of at least 9 miles in less than 1 hour. If John jogs j miles and rides his bike b miles, which of the following systems of inequalities represents John's goal?
Since rate × time = distance, it follows that time is equal to distance divided by rate. John jogs j miles at 6 miles per hour, so the time he jogs is equal to — — = — hours. Similarly, since John rides his bike b miles 6 miles/hour 6 b at 12 miles per hour, the time he rides his bike is —12 hours. Thus, John's goal to complete his jog and his bike ride in less than 1 hour can be represented by the inequality —+— =<1.Thesystemj+b≥9and—+— <1
Example 6: Maizah bought a pair of pants and a briefcase at a department store. The sum of the prices before sales tax was $130.00. There was no sales tax on the pants and a 9% sales tax on the briefcase. The total Maizah paid, including the sales tax, was $136.75. What was the price, in dollars, of the pants?
Subtracting the sides of the first equation from the corresponding sides of the second equation gives you (P + 1.09B) − (P + B) = 136.75 − 130, which simplifies to 0.09B = 6.75. Now you can divide each side of 0.09B = 6.75 by 0.09.ThisgivesyouВ=— =75.ThisisthevalueofB,theprice,indollars, 0.09 of the briefcase. The question asks for the price, in dollars, of the pants, which is P. You can substitute 75 for B in the equation P + B = 130, which gives you P + 75 = 130, or P = 130 − 75 = 55, so the pants cost $55.
Absolute Value
The absolute value of any real number is nonnegative. An important con sequence of this definition is that |−x| = |x| for any real number x. Another important consequence of this definition is that if a and b are any two real numbers, then |a − b| is equal to the distance between a and b on the number line.
Example 5: The stratosphere is the layer of the Earth's atmosphere that is more than 10 kilometers (km) and less than 50 km above the Earth's surface. Which of the following inequalities describes all possible heights x, in km, above the Earth's surface that are in the stratosphere?
The inequality 10 < x < 50 describes the open interval (10, 50). To describe an interval with an absolute value inequality, use the midpoint and the size The stratosphere is the layer of the Earth's atmosphere that is more than 10 kilometers (km) and less than 50 km above the Earth's surface. Which of the following inequalities describes all possible heights x, in km, above the Earth's surface that are in the stratosphere? A) |x + 10| < 50 B) |x − 10| < 50 C) |x + 30| < 20 D) |x − 30| < 20 236 10 + 50 of the interval. The midpoint of (10, 50) is — = 30. Then observe that 2 the interval (10, 50) consists of all points that are within 20 of the midpoint. That is, (10, 50) consists of x, whose distance from 30 on the number line is less than 20. The distance between x and 30 on the number line is |x − 30|. Therefore, the possible values of x are described by |x − 30| < 20
Linear Equations, Linear Inequalities, and Linear Functions in Context
The practical application of Algebra requires one to define one or more variables which represent quantities in context. One should write one or more expressions, equations, inequalities, or functions that represent the relationship described in context.
Example 13: The graph of line k is shown in the xy-plane above. Which of the following is an equation of a line that is perpendicular to line k?
The slope of line k is 0 − 6 = −2. Since the product of the slopes of 3−0 2x is the slope of the line represented by the equation. y = 1/2x+ 3, is an equation of a line with slope , and thus this line is 22 perpendicular to line k. perpendicular lines is −1, a line that is perpendicular to line k will have slope 1/2. All the choices are in slope-intercept form, and so the coefficient of
Example 2: In 2014, County X had 783 miles of paved roads. Starting in 2015, the county has been building 8 miles of new paved roads each year. At this rate, if n is the number of years after 2014, which of the following functions f gives the number of miles of paved road there will be in County X? (Assume that no paved roads go out of service.)
This question already defines the variable and asks you to create a function that describes the context. The discussion in Example 1 shows that the cor rect answer is choice C.
Example 10: −2x = 4y + 6 2(2y + 3) = 3x − 5 What is the solution (x, y) to the system of equations above?
You can solve quickly by substitution. Since −2x=4y+6,it follows that −x=2y+3. Now you can subsitute −xfor2y+3in the second equation. This gives you 2(−x) = 3x − 5, which simplifies to 5x = 5, or x = 1. Substituting 1 for x in the first equation gives you −2 = 4y + 6, which simplifies to 4y = −8, or y = −2. Therefore, the solution to the system is (1, −2).
Systems of Linear Equations and Inequalities in Context
You may need to define more than one variable and create more than one equation or inequality to represent a context and answer a question. There are questions on the SAT Math Test that require you to create and solve a system of equations or create a system of inequalities.
The Relationships among Linear Equations, Lines in the Coordinate Plane, and the Contexts They Describe
A system of two linear equations in two variables can be solved by graphing the lines in the coordinate plane. For example, you can graph the system of equations in Example 10 in the xy-plane The point of intersection gives the solution to the system. If the equations in a system of two linear equations in two variables are graphed, each graph will be a line. There are three possibilities: 1. The lines intersect in one point. In this case, the system has a unique solution. 2. The lines are parallel. In this case, the system has no solution. 3. The lines are identical. In this case, every point on the line is a solution, and so the system has infinitely many solutions. By putting the equations in the system into slope-intercept form, the second and third cases can be identified. If the lines have the same slope and differ ent y-intercepts, they are parallel; if both the slope and the y-intercept are the same, the lines are identical. How are the second and third cases represented algebraically? Examples 11 and 12 concern this question.
Fluency in Solving Linear Equations, Linear Inequalities, and Systems of Linear Equations
Creating linear equations, linear inequalities, and systems of linear equa tions that represent a context is a key skill for success in college and careers. It is also essential to be able to fluently solve linear equations, linear inequal ities, and systems of linear equations. Some of the questions in the Heart of Algebra section of the SAT Math Test present equations, inequalities, or systems without a context and directly assess your fluency in solving them. Some fluency questions allow the use of a calculator; other questions do not permit the use of a calculator and test your ability to solve equations, inequalities, and systems of equations by hand. Even for questions where a calculator is allowed, you may be able to answer the question more quickly without using a calculator, such as in Example 9. Part of what the SAT Math Test assesses is your ability to decide when using a calculator to answer a question is appropriate. Example 8 is an example of a question that could appear on either the calculator or no-calculator portion of the Math Test.
Example 8: 3(1 − y)= —3 + 15y 25 What is the solution to the equation above?
Expand left hand side of equation. Multiply each side by 10. Clear the denominators using LCM. Answer is 1/20.
Example 1: In 2014, County X had 783 miles of paved roads. Starting in 2015, the county has been building 8 miles of new paved roads each year. At this rate, how many miles of paved roads will County X have in 2030?
Firstly one must identify the value they are solving for and assign it a variable. In this case the quantity being solved for is the number of roads will will be assigned variable n. The prompt states that County X had 783 miles of road in 2014 making that the starting point which in mathematical terms is the y intercept. The rate of 8 miles a year which is the slope. There are 16 years between 2014 and 2030 which makes the amount of time known as the x value 16. This all makes the equation for solving this n=16*8+783. In 2030 at this rate there will be 911 roads in County X.
Example 14: A voter registration drive was held in Town Y. The number of voters, V, registered T days after the drive began can be estimated by the equation V = 3,450 + 65T. What is the best interpretation of the number 65 in this equation?
For each day that passes, it is the next day of the registration drive, and so T increases by 1. When T increases by 1, the value of V = 3,450 + 65T increases by 65. That is, the number of voters regis tered increased by 65 for each day of the drive. Therefore, 65 is the number of voters registered each day during the drive.
Example 12: 3s − 2t = a −15s + bt = −7 In the system of equations above, a and b are constants. If the system has infinitely many solutions, what is the value of a?
If a system of two linear equations in two variables has infinitely many solutions, the two equations in the system must be equivalent. Since the two equations are presented in the same form, the second equation must be equal to the first equation multiplied by a constant. Since the coefficient of s in the second equation is −5 times the coefficient of s in the first equa tion, multiply each side of the first equation by −5. This gives you the system −15s + 10t = −5a −15s + bt = −7 Since these two equations are equivalent and have the same coefficient of s, the coefficients of t and the constants on the right-hand side must also be the same. Thus, b = 10 and −5a = −7. Therefore, the value of a is 7/5
Example 11: 2y + 6x = 3 y + 3x = 2 How many solutions (x, y) are there to the system of equations above?
If you multiply each side of y + 3x = 2 by 2, you get 2y + 6x = 4. Then subtracting each side of 2y + 6x = 3 from the corresponding side of 2y + 6x = 4 gives 0 = 1. This is a false statement. Therefore, the system has zero solutions (x, y).
Example 3: In 2014, County X had 783 miles of paved roads. Starting in 2015, the county has been building 8 miles of new paved roads each year. At this rate, in which year will County X first have at least 1,000 miles of paved roads? (Assume that no paved roads go out of service.)
the expression 783 + 8n gives the num ber of miles of paved roads in County X. The question is asking when there will first be at least 1,000 miles of paved roads in County X. This condition can be represented by the inequality 783 + 8n ≥ 1,000. To find the year in which there will first be at least 1,000 miles of paved roads, you solve this inequality for n. Subtracting 783 from each side of 783 + 8n ≥ 1,000 gives 8n ≥ 217. Then dividing each side of 8n ≥ 217 gives n ≥ 27.125. Note that an important part of relating the inequality 783 + 8n ≥ 1,000 back to the con text is to notice that n is counting calendar years and so it must be an integer. The least value of n that satisfies 783 + 8n ≥ 1,000 is 27.125, but the year 2014 + 27.125 = 2041.125 does not make sense as an answer, and in 2041, there would be only 783 + 8(27) = 999 miles of paved roads in the county. Therefore, the variable n needs to be rounded up to the next integer, and so the least possible value of n is 28. Therefore, the year that County X will first have at least 1,000 miles of paved roads is 28 years after 2014, or 2042.
Example 9: −2(3x − 2.4) = −3(3x − 2.4) What is the solution to the equation above?
the structure of the equation reveals that −2 times a quantity, 3x − 2.4, is equal to −3 times the same quantity. This is only pos sible if the quantity 3x − 2.4 is equal to zero. Thus, 3x − 2.4 = 0, or 3x = 2.4. Therefore, the solution is x = 0.8.
