Question

find the slope and y-intercept of the equation

1. ( y = 3x + 2 )
2. ( 3y = - 6x + 9 )
3. ( y = 3 )
4. ( 2y = 4x - 8 )

find the value of the following function for a given variable (place the given value at the variable place and simplify).
14 ) ( f(x) = 5(3 - 2x) ) for ( x = 2 )

1. ( f(x) = x^2 - 5x ) for ( x = 4 )

1. solve inequality. ( 2x + 7 < 5 )
2. check if the given number 5 is a solution of the following inequality ( 3 + 2y < 12 )

show the work and write yes or no on answer sheet.

solve each system of linear equations. write your answer as ordered pair of x and y for # 18,19.

1. ( \begin{cases}x - y = 3 \\ x + y = 5end{cases} )
2. ( \begin{cases}3x + y = 5 \\ x - 2y = 4end{cases} )

Explanation:

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Problems 10-13: Slope and y-intercept

Step1: Use slope-intercept form $y=mx+b$

Here, $m$ = slope, $b$ = y-intercept.

Problem 10: $y=3x+2$

Step1: Match to $y=mx+b$

$m=3$, $b=2$

Problem 11: $3y=-6x+9$

Step1: Isolate $y$

$\frac{3y}{3}=\frac{-6x+9}{3} \implies y=-2x+3$

Step2: Match to $y=mx+b$

$m=-2$, $b=3$

Problem 12: $y=3$

Step1: Rewrite as $y=0x+3$

Step2: Match to $y=mx+b$

$m=0$, $b=3$

Problem 13: $2y=4x-8$

Step1: Isolate $y$

$\frac{2y}{2}=\frac{4x-8}{2} \implies y=2x-4$

Step2: Match to $y=mx+b$

$m=2$, $b=-4$
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Problems 14-15: Evaluate functions

Problem 14: $f(x)=5(3-2x)$ for $x=2$

Step1: Substitute $x=2$

$f(2)=5(3-2(2))$

Step2: Simplify inside parentheses

$5(3-4)=5(-1)$

Step3: Calculate final value

$-5$

Problem 15: $f(x)=x^2-5x$ for $x=4$

Step1: Substitute $x=4$

$f(4)=(4)^2-5(4)$

Step2: Compute each term

$16-20$

Step3: Calculate final value

$-4$
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Problem 16: Solve inequality $2x+7<5$

Step1: Subtract 7 from both sides

$2x+7-7<5-7 \implies 2x<-2$

Step2: Divide by 2

$\frac{2x}{2}<\frac{-2}{2} \implies x<-1$
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Problem 17: Check if $y=5$ solves $3+2y<12$

Step1: Substitute $y=5$

$3+2(5)<12$

Step2: Simplify left side

$3+10=13$

Step3: Compare inequality

$13<12$ is false
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Problems 18-19: Solve linear systems

Problem 18:

$$\begin{cases} x-y=3 \\ x+y=5 \end{cases}$$

Step1: Add the two equations

$(x-y)+(x+y)=3+5 \implies 2x=8$

Step2: Solve for $x$

$\frac{2x}{2}=\frac{8}{2} \implies x=4$

Step3: Substitute $x=4$ into $x+y=5$

$4+y=5 \implies y=1$

Problem 19:

$$\begin{cases} 3x+y=5 \\ x-2y=4 \end{cases}$$

Step1: Isolate $y$ from first equation

$y=5-3x$

Step2: Substitute into second equation

$x-2(5-3x)=4$

Step3: Simplify and solve for $x$

$x-10+6x=4 \implies 7x=14 \implies x=2$

Step4: Substitute $x=2$ into $y=5-3x$

$y=5-3(2)=5-6=-1$
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Answer:

1. Slope: $3$, y-intercept: $2$
2. Slope: $-2$, y-intercept: $3$
3. Slope: $0$, y-intercept: $3$
4. Slope: $2$, y-intercept: $-4$
5. $-5$
6. $-4$
7. $x<-1$
8. No
9. $(4, 1)$
10. $(2, -1)$