Find the best solutions to your problems with the help of IDNLearn.com. Our platform offers reliable and detailed answers, ensuring you have the information you need.

The system of linear equations [tex]\( 3x + 2y = -4 \)[/tex] and [tex]\( y = \frac{1}{2}x + 1 \)[/tex] is graphed on a coordinate plane. Approximate the solution to the system.

A. [tex]\((-1.5, 0.25)\)[/tex]
B. [tex]\((-0.5, 0.7)\)[/tex]
C. [tex]\((1.5, 0.25)\)[/tex]
D. [tex]\((-2.5, 1.5)\)[/tex]


Sagot :

Let's solve the system of linear equations given by:
[tex]\[ \text{Equation 1:} \quad 3x + 2y = -4 \][/tex]
and
[tex]\[ \text{Equation 2:} \quad y = \frac{1}{2}x + 1 \][/tex]

We need to find the point that satisfies both equations.

### Verify the given points

#### Point 1: [tex]\((-1.5, 0.25)\)[/tex]

Substitute [tex]\((-1.5, 0.25)\)[/tex] into Equation 1:
[tex]\[ 3(-1.5) + 2(0.25) = -4 \][/tex]
[tex]\[ -4.5 + 0.5 = -4 \][/tex]
[tex]\[ -4 = -4 \quad \text{(True)} \][/tex]

Substitute [tex]\((-1.5, 0.25)\)[/tex] into Equation 2:
[tex]\[ 0.25 = \frac{1}{2}(-1.5) + 1 \][/tex]
[tex]\[ 0.25 = -0.75 + 1 \][/tex]
[tex]\[ 0.25 = 0.25 \quad \text{(True)} \][/tex]

Since [tex]\((-1.5, 0.25)\)[/tex] satisfies both equations, this point is a solution to the system of equations.

#### Point 2: [tex]\((-0.5, 0.75)\)[/tex]

Substitute [tex]\((-0.5, 0.75)\)[/tex] into Equation 1:
[tex]\[ 3(-0.5) + 2(0.75) = -4 \][/tex]
[tex]\[ -1.5 + 1.5 = -4 \][/tex]
[tex]\[ 0 = -4 \quad \text{(False)} \][/tex]

Since [tex]\((-0.5, 0.75)\)[/tex] does not satisfy Equation 1, it cannot be a solution to the system.

#### Point 3: [tex]\((1.5, 0.25)\)[/tex]

Substitute [tex]\((1.5, 0.25)\)[/tex] into Equation 1:
[tex]\[ 3(1.5) + 2(0.25) = -4 \][/tex]
[tex]\[ 4.5 + 0.5 = -4 \][/tex]
[tex]\[ 5 = -4 \quad \text{(False)} \][/tex]

Since [tex]\((1.5, 0.25)\)[/tex] does not satisfy Equation 1, it cannot be a solution to the system.

#### Point 4: [tex]\((-2.5, 1.5)\)[/tex]

Substitute [tex]\((-2.5, 1.5)\)[/tex] into Equation 1:
[tex]\[ 3(-2.5) + 2(1.5) = -4 \][/tex]
[tex]\[ -7.5 + 3 = -4 \][/tex]
[tex]\[ -4.5 = -4 \quad \text{(False)} \][/tex]

Since [tex]\((-2.5, 1.5)\)[/tex] does not satisfy Equation 1, it cannot be a solution to the system.

### Conclusion
The only point that satisfies both given equations is [tex]\((-1.5, 0.25)\)[/tex]. Thus, the solution to the system is [tex]\(\boxed{(-1.5, 0.25)}\)[/tex].