Question

parallel or perpendicular

1. write the equation of the line that passes through the point (4,2) and is perpendicular to y = 2x - 6

equation:
on to the system?

1. \\(\

$$\begin{cases} y = \\frac{1}{3}x + 2 \\\\ x + y = -2 \\end{cases}$$

\\)
solution:

1. \\(\

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

\\)
solution:
exponents

1. \\(2(3x^{34})(-x^{25})\\)

answer:

1. \\((3x^3y^2)(-x^5y)\\)

answer:

Explanation:

Step1: Find perpendicular slope

The slope of $y=2x-6$ is $2$. Perpendicular slope is $m=-\frac{1}{2}$.

Step2: Use point-slope form

Point $(4,2)$, formula $y-y_1=m(x-x_1)$
$y-2=-\frac{1}{2}(x-4)$

Step3: Simplify to slope-intercept

$y-2=-\frac{1}{2}x+2$
$y=-\frac{1}{2}x+4$

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Step1: Substitute $y$ into linear equation

Substitute $y=\frac{1}{3}x+2$ into $x+y=-2$:
$x+\frac{1}{3}x+2=-2$

Step2: Solve for $x$

$\frac{4}{3}x=-4$
$x=-3$

Step3: Solve for $y$

$y=\frac{1}{3}(-3)+2=-1+2=1$

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Step1: Substitute $y$ into linear equation

Substitute $y=x+5$ into $x-y=2$:
$x-(x+5)=2$

Step2: Simplify and check consistency

$x-x-5=2$
$-5=2$ (false, no solution)

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Step1: Multiply coefficients first

$2\times3\times(-1)=-6$

Step2: Add exponents of $x$

$x^{34+25}=x^{59}$

Step3: Combine terms

$-6x^{59}$

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Step1: Multiply coefficients first

$3\times(-1)=-3$

Step2: Add exponents of $x$

$x^{3+5}=x^8$

Step3: Add exponents of $y$

$y^{2+1}=y^3$

Step4: Combine terms

$-3x^8y^3$

Answer:

1. $y=-\frac{1}{2}x+4$
2. $(-3, 1)$
3. No solution
4. $-6x^{59}$
5. $-3x^8y^3$