QuestionFind the intersection point of lines $y-x-12=0$ and $2x+4y=6$ . Choose from: A $(3,4)$ B $(7,5)$ C $(-3,8)$ D $(7,-5)$ I $(5,-7)$ .

Studdy Solution

STEP 1

Assumptions1. The lines $y=x+12$ and $y=-\frac{1}{}x+\frac{3}{}$ intersect at point $. We need to find the coordinates of point$

STEP 2

The point of intersection of two lines is the solution to the system of equations formed by the two lines. So, we need to solve the system of equations $\begin{aligned} y & =x+12 \\ y & =-\frac{1}{2}x+\frac{}{2} \end{aligned}$

STEP 3

To find the x-coordinate of the point of intersection, we set the two equations equal to each other and solve for $x$ .
$x+12 = -\frac{1}{2}x+\frac{3}{2}$

STEP 4

To solve this equation, we first get rid of the fractions by multiplying every term by2.
$2x+24 = -x+3$

STEP 5

Next, we add $x$ to both sides of the equation to get all $x$ terms on one side.
$2x+x+24 =3$

STEP 6

implify the left side of the equation.
$3x+24 =3$

STEP 7

Subtract24 from both sides of the equation to isolate $3x$ .
$3x =3 -24$

STEP 8

implify the right side of the equation.
$3x = -21$

STEP 9

Divide both sides of the equation by3 to solve for $x$ .
$x = -21 /3$

STEP 10

implify the right side of the equation to find the x-coordinate of the point of intersection.
$x = -7$

STEP 11

Now that we have the x-coordinate, we can find the y-coordinate by substituting $x = -7$ into either of the original equations. Let's use the first equation $y = x+$ .
$y = -7 +$

STEP 12

implify the right side of the equation to find the y-coordinate of the point of intersection.
$y =5$ So, the point of intersection of the two lines is $(-7,5)$ . Looking at the options, we see that this corresponds to option C $(-7,5)$ .

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QuestionFind the intersection point of lines $y-x-12=0$ and $2x+4y=6$ . Choose from: A $(3,4)$ B $(7,5)$ C $(-3,8)$ D $(7,-5)$ I $(5,-7)$ .

Studdy Solution

STEP 1

Assumptions1. The lines $y=x+12$ and $y=-\frac{1}{}x+\frac{3}{}$ intersect at point $. We need to find the coordinates of point$

STEP 2

The point of intersection of two lines is the solution to the system of equations formed by the two lines. So, we need to solve the system of equations $\begin{aligned} y & =x+12 \\ y & =-\frac{1}{2}x+\frac{}{2} \end{aligned}$

STEP 3

To find the x-coordinate of the point of intersection, we set the two equations equal to each other and solve for $x$ .
$x+12 = -\frac{1}{2}x+\frac{3}{2}$

STEP 4

To solve this equation, we first get rid of the fractions by multiplying every term by2.
$2x+24 = -x+3$

STEP 5

Next, we add $x$ to both sides of the equation to get all $x$ terms on one side.
$2x+x+24 =3$

STEP 6

implify the left side of the equation.
$3x+24 =3$

STEP 7

Subtract24 from both sides of the equation to isolate $3x$ .
$3x =3 -24$

STEP 8

implify the right side of the equation.
$3x = -21$

STEP 9

Divide both sides of the equation by3 to solve for $x$ .
$x = -21 /3$

STEP 10

implify the right side of the equation to find the x-coordinate of the point of intersection.
$x = -7$

STEP 11

Now that we have the x-coordinate, we can find the y-coordinate by substituting $x = -7$ into either of the original equations. Let's use the first equation $y = x+$ .
$y = -7 +$

STEP 12

implify the right side of the equation to find the y-coordinate of the point of intersection.
$y =5$ So, the point of intersection of the two lines is $(-7,5)$ . Looking at the options, we see that this corresponds to option C $(-7,5)$ .

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QuestionFind the intersection point of lines y−x−12=0y-x-12=0y−x−12=0 and 2x+4y=62x+4y=62x+4y=6. Choose from: A (3,4)(3,4)(3,4) B (7,5)(7,5)(7,5) C (−3,8)(-3,8)(−3,8) D (7,−5)(7,-5)(7,−5) I (5,−7)(5,-7)(5,−7).

Studdy Solution

STEP 1

Assumptions1. The lines y=x+12y=x+12y=x+12 and y=−1x+3y=-\frac{1}{}x+\frac{3}{}y=−1​x+3​ intersect at point .Weneedtofindthecoordinatesofpoint . We need to find the coordinates of point .Weneedtofindthecoordinatesofpoint

STEP 2

The point of intersection of two lines is the solution to the system of equations formed by the two lines. So, we need to solve the system of equationsy=x+12y=−12x+2\begin{aligned} y & =x+12 \\ y & =-\frac{1}{2}x+\frac{}{2} \end{aligned} yy​=x+12=−21​x+2​​

STEP 3

To find the x-coordinate of the point of intersection, we set the two equations equal to each other and solve for xxx.x+12=−12x+32x+12 = -\frac{1}{2}x+\frac{3}{2}x+12=−21​x+23​

STEP 4

To solve this equation, we first get rid of the fractions by multiplying every term by2.2x+24=−x+32x+24 = -x+32x+24=−x+3

STEP 5

Next, we add xxx to both sides of the equation to get all xxx terms on one side.2x+x+24=32x+x+24 =32x+x+24=3

STEP 6

implify the left side of the equation.3x+24=33x+24 =33x+24=3

STEP 7

Subtract24 from both sides of the equation to isolate 3x3x3x.3x=3−243x =3 -243x=3−24

STEP 8

implify the right side of the equation.3x=−213x = -213x=−21

STEP 9

Divide both sides of the equation by3 to solve for xxx.x=−21/3x = -21 /3x=−21/3

STEP 10

implify the right side of the equation to find the x-coordinate of the point of intersection.x=−7x = -7x=−7

STEP 11

Now that we have the x-coordinate, we can find the y-coordinate by substituting x=−7x = -7x=−7 into either of the original equations. Let's use the first equation y=x+y = x+y=x+.y=−7+y = -7 +y=−7+

STEP 12

implify the right side of the equation to find the y-coordinate of the point of intersection.y=5y =5y=5So, the point of intersection of the two lines is (−7,5)(-7,5)(−7,5). Looking at the options, we see that this corresponds to option C (−7,5)(-7,5)(−7,5).

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QuestionFind the intersection point of lines $y-x-12=0$ and $2x+4y=6$ . Choose from: A $(3,4)$ B $(7,5)$ C $(-3,8)$ D $(7,-5)$ I $(5,-7)$ .

Assumptions1. The lines $y=x+12$ and $y=-\frac{1}{}x+\frac{3}{}$ intersect at point $. We need to find the coordinates of point$

The point of intersection of two lines is the solution to the system of equations formed by the two lines. So, we need to solve the system of equations $\begin{aligned} y & =x+12 \\ y & =-\frac{1}{2}x+\frac{}{2} \end{aligned}$

To find the x-coordinate of the point of intersection, we set the two equations equal to each other and solve for $x$ .
$x+12 = -\frac{1}{2}x+\frac{3}{2}$

To solve this equation, we first get rid of the fractions by multiplying every term by2.
$2x+24 = -x+3$

Next, we add $x$ to both sides of the equation to get all $x$ terms on one side.
$2x+x+24 =3$

implify the left side of the equation.
$3x+24 =3$

Subtract24 from both sides of the equation to isolate $3x$ .
$3x =3 -24$

implify the right side of the equation.
$3x = -21$

Divide both sides of the equation by3 to solve for $x$ .
$x = -21 /3$

implify the right side of the equation to find the x-coordinate of the point of intersection.
$x = -7$

Now that we have the x-coordinate, we can find the y-coordinate by substituting $x = -7$ into either of the original equations. Let's use the first equation $y = x+$ .
$y = -7 +$

implify the right side of the equation to find the y-coordinate of the point of intersection.
$y =5$ So, the point of intersection of the two lines is $(-7,5)$ . Looking at the options, we see that this corresponds to option C $(-7,5)$ .