Algebra (REVIEW)

$1,500 is invested in two different accounts paying 4% and 5% interest. If a total of $67 interest is earned after one year, then how much money was invested at 4%?

$700

The lines whose equations are 2x + y = 3z and x + y = 6z intersect at which point?

(-3z, 9z)

If a boat that travels 23 miles per hour in still water is traveling with a current that has a rate of 2 miles per hour, how far will the boat travel in 3 hours?

75 miles

Solution of System Sets:

A system of two linear equations is classified by the number of ordered pairs that satisfy both equations. Since the graph of each linear equation is a line, three possible situations can occur: no solution, one solution, or an infinite number of solutions. 1. Replace < , >, ≤, or ≥ by = to find the boundary line. 2. Make the line dashed when < or >, and make the line solid when ≤ or ≥. 3. Shade the half-plane whose points satisfy the original inequality. All the points on one side or the other will satisfy the inequality.

John is twice as old as Mary. The sum of their ages is 52. How old is Mary? If J = John's age and M = Mary's age, which system of equations could be used to solve the problem?

J = 2M and J + M = 52

Elimination Method:

The addition property allows us to add (or subtract) two equations. To eliminate a variable, the coefficients must be opposites. The multiplication property of equality allows us to rewrite the equations in order to form opposite coefficients on one of the variables. It is important to line up like terms when adding equations; writing them in standard form will ensure this. Example: https://internationalvla.sooschools.com/media/g_alg01_ccss_2016/5/al203l10_comic01.jpg

linear inequality

an open sentence of the form Ax + By + C < 0 or Ax + By + C > 0

The system shown is _____. https://internationalvla.sooschools.com/media/g_alg01_ccss_2016/5/m90920d.gif

consistent

consistent equations

equations having a common solution in a system

equivalent equations

equations having all common solutions

inconsistent equations

equations having no common solutions in a system

Graph the solution for the following linear inequality system. Click on the graph until the final result is displayed. y > 5 y ≥ x

https://internationalvla.sooschools.com/media/g_alg01_ccss_2016/5/m90917c.gif

https://internationalvla.sooschools.com/media/g_alg01_ccss_2016/5/m90921a.gif Click on the solution set graphic until the correct one is displayed.

infinite set of points on a line

x-determinant

the determinant found when column 1 consists of the constants and column 2 consists of the y-coefficients of a linear system

y-determinant

the determinant found when column 1 consists of the x-coefficients and column 2 consists of the constants of a linear system

system determinant

the determinant found when column 1 consists of the x-coefficients and column 2 consists of the y-coefficients of a linear system

substitute

to replace a quantity with its equal

Given the system of equations, what is the value of the system determinant? 2x + y = 8 x - y = 10

-1

Solve the following system of equations by the substitution method. 5x = y + 6 2x - 3y = 4 What is the value of the y-coordinate?

-8/13

Solving Systems Algebraically:

1. Solve one of the equations for a variable. 2. Substitute the equivalent expression for the variable in step 1 into the other equation. 3. Solve the resulting equation for the other variable. 4. Substitute that value into the one of the original equations. 5. Solve for the other variable. 6. Check the ordered pair in the other equation. 7. State the solution set. Example: Solve the following system of equations by the substitution method. 2x + 6y + 3 = 0 x - 4y - 9 = 0 Solution: You may begin by solving for either variable in either equation. However, since the coefficient of x is 1 in the second equation, this variable will be the best choice to start with. Solve the second equation for x: x - 4y - 9 = 0 x = 4y + 9 Substitute this result, (4y + 9), in the first equation in place of x and find the value of y: 2x + 6y + 3 = 0 2(4y + 9) + 6y + 3 = 0 8y + 18 + 6y + 3 = 0 14y + 21 = 0 14 y = -21 y = - 21/14 y = - 3/2 Now substitute this value of y in either of the original equations to find the value of x: x - 4y - 9 = 0 x - 4(-3/2) - 9 = 0 x + 6 - 9 =0 x - 3 = 0 x = 3

Which of the following equations is equivalent to y = 2/3x + 1/4?

12y = 8x + 3

Determinants:

2x + 6y = -3 (1)x - 4y = 9 The system determinant is d = -14 The x-determinant is d(x) = -42 The y-determinant is d(y) = 21