A theater is installing a straight safety cable on a stage floor. On a coordinate grid, a painted guideline is modeled by the line . The safety cable must be installed perpendicular to this guideline and must pass through the point . Which equation represents the safety cable?

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SAT Linear Equations in Two Variables Question 20 | Free SAT Prep | Aniko

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Question 20·Medium·Linear Equations in Two Variables

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A theater is installing a straight safety cable on a stage floor. On a coordinate grid, a painted guideline is modeled by the line $2x - y = 5$ . The safety cable must be installed perpendicular to this guideline and must pass through the point $(3,-1)$ .

Which equation represents the safety cable?

When a problem involves perpendicular lines, first rewrite the given line as $y=mx+b$ to read the slope $m$ . Then use the negative reciprocal $-\frac{1}{m}$ for the perpendicular slope and plug the given point into point-slope form. Finally, rearrange into the form used by the answer choices to avoid sign mistakes.

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Get the slope from the given equation

Rewrite $2x-y=5$ in the form $y=mx+b$ to identify the slope $m$ .

Use the perpendicular slope rule

If one line has slope $m$ , a perpendicular line has slope $-\frac{1}{m}$ (negative reciprocal).

Use point-slope form

With slope $m$ and point $(x_1,y_1)$ , start from $y-y_1=m(x-x_1)$ .

Match the answer choices

After you get an equation, rewrite it into the same form as the choices (often $Ax+By=C$ ).

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Graph the guideline and plot the point

Enter $2x-y=5$ and add the point $A=(3,-1)$ .

Graph a perpendicular line through the point

Since the guideline has slope $2$ , enter the perpendicular line through $A$ :

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

Match to a choice

Rewrite the line you graphed into standard form and select the answer choice that matches it.

Step-by-step Explanation

Find the slope of the given line

Rewrite $2x-y=5$ in slope-intercept form:

2x-y=5 \Rightarrow -y=5-2x \Rightarrow y=2x-5

So the slope of the guideline is $2$ .

Use the perpendicular slope and the given point

A line perpendicular to slope $2$ has slope $-\frac{1}{2}$ .

Use point-slope form with point $(3,-1)$ :

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

y+1=-\frac{1}{2}x+\frac{3}{2}

Convert to standard form and match a choice

Subtract 1 from both sides:

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

Multiply by 2:

2y=-x+1 \Rightarrow x+2y=1

So the correct choice is $x + 2y = 1$ .

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Step-by-step Explanation

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