Banneker

Welcome to Banneker's wiki-space. It has been a great pleasure creating our page on Solving Equations for "y". Below we have examples for learning how to work out solving for y and for solving word problems for y. In red are important formulas to know and in bold are our main topics, the beginning & final answer and also we have labels of which level math class that lesson is taught in. Have fun exploring. :)

**//__Pre-Algebra:__ //**
** //Slope-Intercept Form//- **

Solve for Slope format
 * y = mx + b **

**3x + 4y = 12** // 1.Subtract 3x from each side so you get 4y = 12 - 3x 2.Divide each side by 4 You get **y = -4/3 + 3** Congratulations you have now solved for y //

**Problem Solving:** You buy vegetables at a farmer’s market for $2 per pound. The equation y=2x represents the situation where x is the number of pounds and y is the total cost of your purchase. Graph the equation. Y=2x Slope-intercept form= y=mx+b M=2 B=0 Step 1: The y-intercept is 0 so plot the point at (0,0) Step 2: The slope is 2, or 2/1. Use the slope to plot a second point Step 3: Draw a line through the two points.

**__// Algebra 1: //__ ​**

 * // Linear Equations- //**

Graph - //1. Add 2 to each side So you get// ** //** y = x - 1 **// ** //2. Now you can graph that by...// ** media type="file" key="how to solve a line.mov" width="679" height="679" ​ **// __ Point Slope Form- __ //**
 * y - 2 = ( x - 3)

//** y1 - y = m ( x1 - x ) **// ​

Put in point-slope form- //1. Substitute the values into the formula 2. So you get y1-(-7)= -3/2[x-(-2)] 3. Simplify y1+7= -3/2(x+2)//
 * (-2,-7); m= -3/2 **

Put in point-slope form then in slope intercept form- ** ( 6, -4 ) , ( -3 , 5 ) ** //1. Substitute the values into the formula 5-(-4) = m (-3-6) 2. Simplify 9=m(-9) 3. Divide both sides by negative 9 and you get M= -1 4. Subsitute M iand the first ordered pait into the point slope form and you will get a slope intercept equation for the problem :)//

**Problem Solving:** An escalator has a slope of 2. After traveling forward 12 feet, the escalator is 24 feet above the floor. Put this equation into point-slope form, then slope-intercept form. Slope=2 (12,24) (y-y^1)=m(x-x^1) (y-24)=2(x)-2(12) (y-24+24)=2x-24+24
 * (y-24)=2(x-12) - Point-slope form **
 * Y=2x - Slope- Intercept Form **

** // __Standard Form__ //// __-__ // **

//** Ax + By = C **//

Find x & y- **-6x + 3y = -9** //1. add 6x to each side you get 3y=6x-9 2. divide everything by 3 You end up geting ** y = 2x - 3 ** From there you can graph it.//

Write an equation in standard form to find the number of minutes someone who weighs 150 lb would need to bicycle and swim laps in order to burn 300 calories. Then solve to find the x and y intercepts. __Activity by 150 lb person Calories Burned per Minute__ Bicycling 10 Bowling 4 Hiking 7 Running 5.2 mi/h 11 Swimming laps 12 Walking 3.5 mi/h 5 Define: Let x= the minutes spent bicycling Let y= the minutes spent swimming laps Relate: 10* minutes bicycling + 12*minutes swimming laps =300 calories Write: 10x+12y =300 The equation in standard form is 10x +12y =300 Finding the x and y intercepts: 10x+12y=300 10(0) +12y=300 12y=300 12y/12= 300/12 Y= 25 10x+12y=300 10x+12(0)=300 10x/10=300/10 X=30
 * Problem Solving: **
 * They would need to bicycle for 30 minutes and swim laps for 25 minutes. **

// ** Multi-Step Equations ** - //

Solve and check-

1 Step- //1. multiply each side by 4 You get -3y = 36 2.divide each side by -3 You get ** y = -12 **//
 * -3/4 y = 9 **

2 Step- //** 4.2 y + 4 = 25 ** 1.Subtract 4 from side You get 4.2y=21 2.Divide each side by 4.2 You get ** y = 5 **//

3+ Step- // 1. Combine like terms You get 10y + 7= -33 2. Subtract 7 from each side You get 10y = 26 3. Divide each side by 10 You end up with // // y = 2.6 // **
 * 4y + 7 + 6y = -33

You can buy used in-line skates from your friend for $40, or you can rent some($3.50/hour with safety equipment). Either way you must rent safety equipment ($1.50/hour). How many hours must you skate for the cost of renting and buying skates to be the same? Relate: cost of friend’s skates + safety equipment = skates plus equipment rental Define: let y=the number of hours you must skate Write : 40+1.5y=3.5y 40+1.5y=3.5y 40+1.5y-1.5y=3.5y-1.5y 40=2y 40/2=2y/2 20=y
 * Problem Solving: **
 * You must skate 20 hours for the cost to be the same **


 * // Direct and Inverse Variation- //**


 * Direct Variation- **


 * y = kx **

Find the constant variation-
 * y = x - 3 **

Write an equation of direct variation- ** Problem Solving: ** Your distance from lightning varies directly with the time it takes you to heat thunder. If you hear thunder 10 seconds after you see lightning, you are about 2 miles from the lightning. Write an equation for the relationship between thunder and lightning. Relate: The distance varies directly with the time. When x=10, y=2 Define: Let x= the number of seconds between your seeing lightning and your hearing thunder Let y = your distance in miles from the lightning. Write y=kx 2=k(10) 2/10=10k/10 1/5=k Y=1/5x
 * <span style="color: #000080; font-family: 'Times New Roman',Times,serif;">The equation y=1/5x relates the time(x) in seconds it takes you to hear the thunder to the distance(y) in miles you are from the lightning. **


 * Inverse Variation **


 * xy = k or y = k/x **

Write an equation for an inverse variation- //1. Substitute y and x into the inverse variation formula 6(1)= k 2. The constant is six and substitute that into the formula 3. Now you have y= 6/x//
 * x = 6 when y = 1 **

Find the missing value-
 * ( 9, y) and ( 3 , 12 ) **

<span style="color: #000080; font-family: 'Times New Roman',Times,serif;">The number of hours, y, it takes for a block of ice to melt varies inversely as the temperature, //x//. If it takes 2 hours for a square inch of ice to melt at 65º, find the constant of proportionality. Start with the formula: y=k/x Substitute the values : 2=k/65
 * <span style="color: #000080; font-family: 'Times New Roman',Times,serif;">Problem Solving: **


 * then solve for //k//: k=130**

//** Systems; Elimination and Substitution- **//

**Elimination**

Solve -7x + y = -16 ** 1. //Add both equations together equations now you have 3y= -6, since +7x and -7x cancel each other out 2. Divide both sides by 3 and you have y = -2 3. Substitute y into one equation to find x -7x - 2 =10// //4. Add two to both sides -7x - 2 + 2 = 10 +2 and now you have -7x = 12 5. Divide both sides by -7 and you end up with ** x = -12/7 **//
 * 7x + 2y = 10

<span style="color: #000080; font-family: 'Times New Roman',Times,serif;">**Problem Solving:** Suppose your class sells gift wrap for $4 a package and greeting cards for $10 a package. Your class sells 205 packages in all and receives a total of $1084. Find the number of each type of package sold. Define: Let w+ number of packages of gift wrap sold Let c=number of packages of greeting cards sold Relate: Total number of packages total amount of sales Write w+c=205 4w+10c=1084 Step 1: Eliminate one variable W+c=205 __4w+10c=1084__ 4(w+c=205) __4w+10c=1084__ 4w+4c=820 __-(4w+10c=1084)__ 0-6c=-264 Step 2: Solve for c -6c=-264 C=44 Step 3: Solve for the eliminated variable using either of the original equations W+c=205 W+44=205 W=161
 * <span style="color: #000080; font-family: 'Times New Roman',Times,serif;">The class sold 161 packages of gift wrap and 44 packages of greeting cards **

**Substitution**

Solve - h = -2y + 28 ** <span style="font-family: 'Times New Roman',Times,serif;">Problem Solving: ** <span style="color: #000080; font-family: 'Times New Roman',Times,serif;">Your school must transport 193 people to a competition. There are eight drivers available and two types of vehicles. The school buses seat 51 people each, and the minivans seat 8 people each. How many buses and minivans will be needed? Define:Let b=number of buses Let m=number of minivans Number of drivers: b+m=8 Number of people: 51b+8m=193 Step 1: b+m=8 M=-b+8 Step 2: 51b+8(-b+8)=193 51b-8b+64=193 43b+129 B=3 Step 3: b+m=8 3+m=8 M=5
 * h = 6y - 4
 * <span style="color: #000080; font-family: 'Times New Roman',Times,serif;">Three school buses and five minivans will be needed to transport 193 people. **


 * // Linear Inequalities- // **

Solve and Check-

//1. Subtract 7 from each side so you get 4y < 16 2. Divide each side by four so you get y < 4 3. Check 4(2)+ 7< 23 8 +7 & < 23 15 < 23//
 * 4y + 7 < 23 **

<span style="color: #000080; font-family: 'Times New Roman',Times,serif;">**Problem Solving:** Suppose your budget for a party allows you to spend no more than $12 on peanuts and cashews. Peanuts cost $2/lb and cashews cost $4/lb. Find three possible combinations of peanuts and cashews you can buy. Relate: cost of peanuts + cost of cashews__<__ total budget Define: Let x+ the number of pounds of peanuts Let y+ the number of pounds of cashews Write: 2x+4y=12 Graph 2x+4y=12 by graphing the intercepts (6,0) and (0,3) The coordinates of points on the boundary line make the inequality true. So, use the solid line. Graph only Quadrant 1, since you cannot buy a negative amount of peanuts or cashews. Test the point (1,1) 2x+4y__<__12 2(1) +4(1)__<__ 12 2+4__<__ 12 6__<__12 Shade the region containing (1,1) The graph below shows all the possible solutions of the problem
 * <span style="color: #000080; font-family: 'Times New Roman',Times,serif;">Since the boundary line is included in the graph, the intercepts are also solutions of the inequality. The solution (2,2) means that if you buy 2 lb of peanuts, you can buy 2 lb of cashews. Three solutions are (2,2), (3,1) and (1,2). **

Below is a video about linear equations.

** //Parallel and Perpendicular Lines-// **
 * Perpendicular Lines**

Find the slope of a line perpendicular to the graph.
 * y = -3x **

Write an equation for the line that is perpendicular to the given line and that passes through the given point.
 * y = 2x + 7 ; ( 0, 0 ) **

<span style="color: #000080; font-family: 'Times New Roman',Times,serif;">A jogging path for a new city park will connect the park (4) entrance to Park Roads The path will be perpendicular to Park Road. Write an equation for the line representing the jogging path. Step 1: Find the slope (m) of park road: points are (2,1) and (4,5) M=(y^2-y^1)/(x^2-x^1) M= (5-1)/(4-2) M=4/2 M=2 Step 2: Find the negative reciprocal of the slope The negative reciprocal of 2 is -1/2. So the slope of the bike path is -1/2
 * <span style="color: #000080; font-family: 'Times New Roman',Times,serif;">Problem Solving: **
 * <span style="color: #000080; font-family: 'Times New Roman',Times,serif;">The equation for the bike path is y=-1/2x+4 **
 * Parallel Lines**

Are the lines parallel? -4x + 3y = 21 **
 * y = 4x + 12

Write an equation for the line that is parallel to the given line and that passes through the given point.
 * y = 6x - 2 ; ( 0, 0 ) **

<span style="color: #000080; font-family: 'Times New Roman',Times,serif;">A second jogging path is planned. It will be parallel to Park Road and will also contain the park entrance (4). Write an equation for the line representing this jogging path.
 * <span style="color: #000080; font-family: 'Times New Roman',Times,serif;">The equation would be y=2x+4 because parallel lines have the same slope and because it goes through the park entrance the y-interc ept is 4.**

//** Standard Form of a Quadratic Function- **


 * y = ax^2 + bx + c **//

Solve for y.
 * r^2 + 8r + 48 = y **

<span style="color: #000080; font-family: 'Times New Roman',Times,serif;">**Problem Solving:** In professional fireworks displays, aerial fireworks carry “stars” upward, ignite them, and project them into the air. Suppose a particular star is projected from an aerial firework at a starting height of 520ft with an initial upward velocity of 72 ft/s. How long will it take for the star to reach its maximum height? How far above the ground will it be? The equation h=-16t^2+72t+520 gives the star’s height (h) in feet at time (t) in seconds. Since the coefficient of t^2 is negative, the curve opens downward, and the vertex is the maximum point. Step 1: Find the coordinate of the vertex -b/2a=-(72)/2(-16)=2.25 After 2.25 seconds, the star will be at its greatest height Step 2: Find the h-coordinate of the vertex H=-16t^2+vt+c H=-16(2.25)^2+72(2.25)+520 H=601
 * <span style="color: #000080; font-family: 'Times New Roman',Times,serif;">The maximum height of the star will be 601 feet **

Make a table of values for the function.
 * //Square Root Functions// **
 * f (x) = 2 V x- 4 **

<span style="color: #000080; font-family: 'Times New Roman',Times,serif;">**Problem Solving:** In good weather conditions, police can use the formula r =2√5L to find the approximate speed (r) of a car that leaves a skid mark of length (l) in feet. Graph the function. Length of Skid Mark (ft) Speed(mi/h) 0 0 10 14.1 20 20 30 24.5 40 28.3

=
** <span style="color: #0000ff; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">Sl__ope:__ A number which is used to indicate the steepness of a line as well as indicating whether the line is tilted uphill or downhill. Slope is indicated by the letter <span style="color: #ff00a7; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">//<span style="color: #0000ff; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">m // <span style="color: #0000ff; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">. ** =====

=

 * <span style="color: #ff00a7; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">Intercept: the value where a function crosses the axes, such as the x and y intercept, in the slope-intercept form, the y intercept is known as 'b' **=====

=

 * <span style="color: #0000ff; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">Linear Equation: an algebraic equation in which the variable quantify or quantities are in the first power and the graph is a straight line e.g. 20= 2(w +4) + 2w <span style="color: #ff00a7; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">Point-Slope Form: is used to find the equation of a line **=====

=

 * <span style="color: #ff00a7; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">Direct Variation: A relationship between two variables in which one is a constant multiple of the other. In particular, when one variable changes the other changes in proportion to the first. **=====

=

 * <span style="color: #0000ff; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">Inverse Variation: A relationship between two variables in which the product is a constant. When one variable increases the other decreases in proportion so that the product is unchanged. **=====

=

 * <span style="color: #0000ff; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">Constant Function: A function of the form <span style="font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">//<span style="color: #0000ff; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">y // <span style="color: #0000ff; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;"> = constant or <span style="font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">//<span style="color: #0000ff; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">f // <span style="color: #0000ff; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">( <span style="font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">//<span style="color: #0000ff; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">x // <span style="color: #0000ff; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">) = constant **=====

=

 * <span style="color: #0000ff; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">Elimination: A method for solving a system of linear equations. You add or subtract the equations to eliminate a variable. ** **<span style="color: #ff00a7; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">Linear Inequality: An inequality that can be written in the form "linear polynomial > linear polynomial" or "linear polynomial > constant". The > sign may be replaced by <, ≤, or ≥. **=====

=

 * <span style="color: #ff00a7; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">Perpendicular Lines: equations with slopes that are negative reciprocals will graph perpendicular to each other **=====