In the system of equations below, c is a constant. \begin{aligned} 2x + y &= 15 \\ cx - 2y… — SAT

In the system of equations below,

c

is a constant.

\begin{aligned} 2x + y &= 15 \\ cx - 2y &= 6 \end{aligned}

If the solution

(x, y)

to the system of equations above lies on the line

y = 3x

in the

xy

-plane, what is the value of the constant

c

Cevap: 8

Cevap

The value of the constant

c

is 8.

Since the solution

(x, y)

lies on the line

y = 3x

, we can substitute

y = 3x

into the first equation,

2x + y = 15

, to get

2x + 3x = 15

, which simplifies to

5x = 15

, yielding

x = 3

. Substituting

x = 3

back into

y = 3x

gives

y = 9

. Thus, the intersection point of the system is

(3, 9)

. Substituting

x = 3

and

y = 9

into the second equation,

cx - 2y = 6

, gives

c(3) - 2(9) = 6

, or

3c - 18 = 6

. Adding 18 to both sides gives

3c = 24

, which simplifies to

c = 8

Adım Adım Çözüm

Substitute the equation of the line

y = 3x

into the first equation of the system,

2x + y = 15

2x + 3x = 15

, which simplifies to

5x = 15

. Solving for

x

gives

x = 3

Since the solution to the system lies on the line

y = 3x

, the coordinates of the solution must satisfy all three equations.

Calculate the corresponding

y

-value using the relation

y = 3x

y = 3(3) = 9

The

y

-coordinate is three times the

x

-coordinate for any point on the line

y = 3x

Substitute the point

(3, 9)

into the second equation of the system,

cx - 2y = 6

c(3) - 2(9) = 6

, which simplifies to

3c - 18 = 6

. Adding 18 to both sides gives

3c = 24

, so

c = 8

Since

(3, 9)

is the solution to the system, it must satisfy the second equation as well.

Anahtar Kavram

Solving systems of linear equations using substitution and finding parameter values based on geometric constraints.

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Konular

Bu konudan daha fazla soru