Linear Equations Cheat Sheet PDF

## **Linear Equations** ### Cheat Sheet #### **Parts of an equation** - `-12x + 4 = 40` - **Coefficient:** the number in front of a variable (multiplying the variable). - **Constant:** a fixed value, or a number on its own. - **Variable:** a letter used to represent an unknown value. #### **S...

## **Linear Equations** ### Cheat Sheet #### **Parts of an equation** - `-12x + 4 = 40` - **Coefficient:** the number in front of a variable (multiplying the variable). - **Constant:** a fixed value, or a number on its own. - **Variable:** a letter used to represent an unknown value. #### **Special cases** - **No solution:** An equation where no value for *x* will make the equation true. - Work ends in a false statement, such as "7 = 5". - Graph shows two parallel lines. - **All real numbers:** An equation where any value for *x* will make the equation true. - Work ends in a true statement, such as "7 = 7". - Graph shows the same line. #### **Multi-step equations** ##### **Steps to solve:** 1. Distribute (if necessary) 2. Combine like terms (if necessary) 3. Collect variables on the same side of the equals sign 4. Collect constants on the same side of the equals sign 5. Isolate the variable with inverse operations 6. Check your answer by plugging it in. #### **Solving Systems by Substitution** 1. Solve one of the equations for "y =", if necessary. 2. Substitute the value for "y" in the second equation. 3. Solve the new equation to find "x". 4. Substitute the value for "x" in either of the original equations to find "y". #### **Solving Systems by Graphing** - To solve a system of equations by graphing, graph both linear equations and find the point of intersection (x, y). ## **Linear Equations - Quick Check** 1. Gym A charges a registration fee of $75 plus $35.75 per month for members. Gym B charges a registration fee of $164 plus $17.95 for members. After how many months would the total cost at Gym A and Gym B to be the same for members? - A. 10 months - B. 5 months - C. 7 months - D. The total cost will never be the same. 2. The graph below shows the number of hours that Rue and Zoe have been working at their jobs, as well as how much money they've earned. Which is a correct conclusion about the information shown in the graph? - F. After 24 hours, Rue and Zoe will have earned the same amount of money. - G. After 4 hours, Zoe will have earned $12 more than Rue. - H. After 4 hours, Rue will have earned $12 more than Zoe. - J. After 4 hours, Rue and Zoe will have earned the same amount of money. 3. Find the value of *x* needed to make the equation below true. - (20*x - 8) - 3 = 54 - A. x = 3.8 - B. x = 4.2 - C. x = 4 - D. x = 3.3 4. The area of the rectangle shown below is 36 square units. Set up and solve an equation to find the value of *x*. - 2.5 * *x* + 1.5 - 4 - F. *x* = 4.4 - G. *x* = 3.45 - H. *x* = 3 - J. *x* = 5 ## **Equations and Inequalities** ### Cheat Sheet #### **Distributive Property** - Multiplies each term inside parentheses by the number outside parentheses. - a(b + c) -> ab + ac - a(b - c) -> ab - ac - Check your signs and follow integer rules! #### **Types of solutions** - **One solution:** Only one value will make the equation true. - Example: 2*x* = 10, *x* = 5. - **No solution:** There is not a value that makes the equation true. - Example: 3*x* + 9 = 3*x* - 6, 9 = -6. - **All real numbers:** Any real number will make the equation true. - Example: 2*x* - 8 = 2*x* - 8, -8 = -8. #### **Solving Inequalities** - Apply the same steps as solving equations, but flip the inequality sign when multiplying or dividing both sides by a negative value. - Solutions to inequalities include a set of values and can be represented on a number line. - *x* > 4 - *x* ≥ 4 - *x* < 4 - *x* ≤ 4 #### **Literal equations** - Equations with more than one variable that can be rearranged and solved for a specified variable using inverse operations. - Example: Solve *A* = *b* *h* for *h*. - *A* / *b* = *h* = *h* ## **Equations and Inequalities - Quick Check** 1. What value of *x* makes the equation 6*x* - 14 + 3(2*x* - 1) = 5 - 2(*x* - 10) true? - A. *x* = 2.3 - B. *x* = 3 - C. *x* = -0.3 - D. *x* = 0.71 2. Broderick needs to solve the formula shown below for *r*. Which correctly describes the first step he should take? - *A* = π*r*² - F. He should subtract π from both sides of the equation. - G. He should take the square root of both sides of the equation. - H. He should divide both sides of the equation by *A*. - J. He should divide both sides of the equation by π. 3. Truman is buying a teddy bear and flowers for his girlfriend. Flowers Galore charges $13 for the teddy bear and $0.75 per flower. Famous Florist charges $9 for the teddy bear and $1.25 per flower. Which inequality represents the number of flowers, *f*, Truman would need to buy to make Flowers Galore the cheaper option? - A. *f* > 8 - B. *f* < 8 - C. *f* > 2 4. D. *f* < 2 4. The cheerleaders at McKamey High are painting rectangular banners for the pep rally. The area of the banner is 49 square feet. The dimensions of the rectangle are shown below. What is the length of the longer side of the banner? - 5*x* - 6 ft - 3.5 ft - F. 14.5 ft² - G. 5.4 ft - H. 14 ft - J. 21 ft ## **Functions and Slope** ### Cheat Sheet #### **Functions** - **Function:** A relationship in which every input (*x*) has exactly one output (*y*). ##### **To check if it's a function:** 1. **Ordered pairs & tables:** Each *x*-value must correspond with exactly one *y*-value. Check for repeating *x*-values. 2. **Equations:** See if any input would result in more than one output. For example, *y*² = *x* could result in ± *y*. 3. **Graphs:** Must pass the "vertical line test", where any vertical line touches the graph at only one point. #### **Slope** - Also called the "rate of change". - **Positive:** Increases left to right. - **Negative:** Decreases left to right. - **Zero:** A horizontal line. - **Undefined:** A vertical line ##### **The formula:** - *y*2 - *y*1 / *x*2 - *x*1 - **Rise over run:** *y*2 - *y*1 / *x*2 - *x*1 ##### **Triangles on the same line have the same slope, and are similar triangles. The ratios of their corresponding sides are equal.** ## **Functions and Slope - Quick Check** 1. The table below shows Vanessa's height in inches for two different years. Which is a correct conclusion about the rate of change shown in the table? - YEAR (*x*) | HEIGHT (*y*) - -------- | -------- - 2000 | 48 inches - 2005 | 54 inches - A. Vanessa grows about 41.7 inches per year. - B. Vanessa grows about 6 inches per year. - C. Vanessa grows about 1.2 inches per year. - D. Vanessa grows about .83 inches per year. 2. Ariel is emptying the water from a 10 gallon cooler. The graph shows the water level in the cooler as she empties it. Which best describes the rate of change shown in the graph? - F. The water level decreases 10 gallons per second. - G. The water level decreases 1 gallon every 2 seconds. - H. The water level decreases 3 gallons every 2 seconds. - J. The water level decreases 2 gallons every 3 seconds. 3. The slope of a graphed line is 2/5. Which of the following triangles could lie on the line? - A. 40/16 - B. 50/20 - C. 15/12 - D. 52/22 4. Which of the following situations does not have the same unit rate as the graph shown? - F. Asher buys gum for $0.20 a piece. - G. A daycare has six workers for every 30 children. - H. Melanie reads 9 pages of her book every 45 minutes. - J. Richie earns $10 every 2 hours to pet sit for his neighbor. ## **Linear Relationships** ### Cheat Sheet #### **Slope-intercept form:** - *y* = *m* *x* + *b* - **Slope:** *m* - **Y-intercept:** *b* ##### **Example 1** - A line with a slope of 8 and a y-intercept of -10 would have an equation of *y* = 8*x* - 10. - An equation of *y* = -3*x* + 5 means the slope of the line is -3 and the y-intercept is 5. ##### **Example 2** - Find the slope by using (*y*2 - *y*1) / (*x*2 - *x*1). - Find the y-intercept by finding the value of *y* when *x* = 0, or where a graphed line crosses the y-axis. - write an equation in slope-intercept form (*y* = *m* *x* + *b*) - **Example 1** - **Slope:** 1/2 - **Y-intercept:** 1 - **Equation:** *y* = 1/2 *x* + 1 - **Example 2** - **Slope:** 4 - **Y-intercept:** -9 - **Equation:** *y* = 4*x* - 9 #### **Linear relationships can be represented verbally, with an equation, with a graph, and with a table.** - **Verbal:** A puppy weighs 2 pounds at birth and gains half a pound each week. - **Equation:** *y* = 0.5*x* + 2 - **Table:** - Weeks (*x*) | Weight (*y*) - -------- | -------- - 0 | 2 - 1 | 2.5 - 2 | 3 - 3 | 3.5 - **Graph:** (graph of y = 0.5x + 2) #### **If an equation is linear, it will be written in the form of *y* = *m* *x* + *b*.** - **Linear:** *y* = 0.75*x* - 11 - **Non-linear:** *y* = 3*x*² + 2 #### **If a graph is linear, it will look like a straight line.** #### **Standard form:** - *Ax* + *By* = *C* - Can be helpful for finding x and y-intercepts. - To find x-intercept (or zeros), set *y* = 0. - To find y-intercept, set *x* = 0. #### **Point-slope form:** - *y* - *y*1 = *m*(*x* - *x*1) - Easily written when given the slope (*m*) and a point (*x*1, *y*1). #### **Graphing linear inequalities:** - Solve the inequality for *y* and use the y-intercept and slope to plot points on the line. Use the inequality sign to determine the type of line to use and where to shade the graph. - *y* ≥ *m*x + *b* - *y* < *m*x + *b* - *y* > *m*x + *b* - *y* ≤ *m*x + *b* ## **Linear Relationships - Quick Check** 1. What of the statements about the graph below is true? - A. The graph is non-linear and has an equation of y = x + 18. - B. The graph is linear and has an equation of y = x + 18. - C. The graph is linear and has an equation of y = -2x + 18. - D. None of the above statements are true. 2. The table below shows the amount that a catering company charges based on the number of people at an event. Which of the following equations shows the relationship between the amount the company charges based on *p*, the number of people at the event? - People (*p*) | Total Charge (*C*) - -------- | -------- - 25 | $400 - 50 | $725 - 75 | $1050 - 100 | $1375 - F. C = 75*p* + 13 - G. C = 75*p* + 13 - H. C = 130 + 75*p* - J. C = 13*p* + 75 3. Which of the following gives an example of an equation that is non-linear? - A. *y*=*x*²-1 - B. *y*=2*x*-1 - C. *y*=*x*/4 - D. *y*=-2*x* 4. Kit charges customers and initial fee plus a certain amount per hour to walk their pets. The table below shows the amount of money that Kit earns at her job based on the number of hours that she works. Which of the following equations represents a scenario where Kit would charge customer a higher hourly rate to walk their pets that what is shown in the table? - Hours | Earnings ($) - -------- | -------- - 0 | 15 - 1 | 22.5 - 2 | 30 - 3 | 37.5 - F. y = 7.25*x* + 20 - G. y = 8*x* + 10 - H. y = 7.5*x* - J. y = -10*x* + 15 ## **Near Functions** ### Cheat Sheet #### **Slope & Rate of change:** - Describes the steepness of a line. - Can be found using the formula (*y*2 - *y*1) / (*x*2 - *x*1). - **Types of slope:** - **Positive:** increases left to right. - **Negative:** decreases left to right. - **Zero:** a horizontal line. - **Undefined:** a vertical line - **Easily written when given the slope (*m*) and a point (*x*1, *y*1).** - *y* - *y*1 = *m*(*x* - *x*1) #### **Slope-intercept form:** - *y* = *m* *x* + *b* - **Slope:** *m* - **Y-intercept:** *b* ##### **Example** - Write an equation of the linear function in slope-intercept form. - **Slope:** -2 - **Y-intercept:** 2 - **Equation:** *y* = -2*x* + 2 #### **Standard form:** - *Ax* + *By* = *C* - Can be helpful for finding x and y-intercepts. - To find x-intercept (or zeros), set *y* = 0. - To find y-intercept, set *x* = 0. #### **Point-slope form:** - *y* - *y*1 = *m*(*x* - *x*1) - Solve the inequality for *y* and use the y-intercept and slope to plot points on the line. Use the inequality sign to determine the type of line to use and where to shade the graph. - *y* ≥ *m*x + *b* - *y* < *m*x + *b* - *y* > *m*x + *b* - *y* ≤ *m*x + *b* ## **Near Functions - Quick Check** 1. Luis’ parents give him *x* dollars for his monthly allowance. Each month, he must pay $35 for his cell phone. One-sixth of the remaining money can be spent on entertainment. Which function can be used to find the amount in dollars Luis can spend on entertainment? - A. *f*(*x*) = (*x* - 35) / 6 - B. *f*(*x*) = 35*x* / 6 - C. *f*(*x*) = 35 / *x* - D. *f*(*x*) = *x* / 6 - 35 2. The table at the right represents some points on the graph of a linear function. Which of the following equations represents the function? - x | y - -------- | -------- - -8 | 123 - -2 | 51 - 1 | 75 - 4 | -21 - 7 | -57 - 9 | -81 - F. *y* = 12*x* + 27 - G. *y* = 27*x* - 12 - H. *y* = -12*x* + 27 - J. *y* = 12*x* + 27 3. Gracie graphed the line 3*x* - 4*y* = -16. Which ordered pair is in the solution set of 3*x* - 4*y* ≥ -16? - A. (-2, 3) - B. (-8, 0) - C. (2, 7) - D. (0, -1) 4. Isabella and her aunts assemble tamales to sell at her uncle's restaurant. The graph shows the linear relationship between the number of minutes they have been working and the number of tamales they have assembled. Which of the following is a true statement about the graph? - F. They can assemble 6.25 tamales per minute. - G. They can assemble 8/3 tamales per minute. - H. The initial number of tamales assembled was 0. - J. It takes them 3 minutes to assemble 50 tamales.