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Which ordered pair is a solution to the system of linear equations?

[tex]\[
\begin{array}{l}
x + 4y = 3 \\
y = -4x - 3
\end{array}
\][/tex]

A. [tex]\((1, 1)\)[/tex]
B. [tex]\((1, -1)\)[/tex]
C. [tex]\((-1, 1)\)[/tex]
D. [tex]\((-1, -1)\)[/tex]


Sagot :

Certainly! Let's solve the given system of linear equations to identify the ordered pair that satisfies both equations.

The system is:
[tex]\[ \begin{cases} x + 4y = 3 \\ y = -4x - 3 \end{cases} \][/tex]

We can solve this system by substitution or elimination, but since the second equation already expresses [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex], substitution is convenient.

### Step 1: Substitute [tex]\( y = -4x - 3 \)[/tex] into [tex]\( x + 4y = 3 \)[/tex]

[tex]\[ x + 4(-4x - 3) = 3 \][/tex]

### Step 2: Simplify the equation

[tex]\[ x + 4(-4x) + 4(-3) = 3 \][/tex]

[tex]\[ x - 16x - 12 = 3 \][/tex]

[tex]\[ -15x - 12 = 3 \][/tex]

### Step 3: Solve for [tex]\( x \)[/tex]

[tex]\[ -15x = 3 + 12 \][/tex]

[tex]\[ -15x = 15 \][/tex]

[tex]\[ x = \frac{15}{-15} \][/tex]

[tex]\[ x = -1 \][/tex]

### Step 4: Substitute [tex]\( x = -1 \)[/tex] back into [tex]\( y = -4x - 3 \)[/tex] to find [tex]\( y \)[/tex]

[tex]\[ y = -4(-1) - 3 \][/tex]

[tex]\[ y = 4 - 3 \][/tex]

[tex]\[ y = 1 \][/tex]

Thus, [tex]\( x = -1 \)[/tex] and [tex]\( y = 1 \)[/tex]. The ordered pair that is a solution to the system is [tex]\( (-1, 1) \)[/tex].

### Verification

To ensure our solution is correct, let's substitute [tex]\( (-1, 1) \)[/tex] back into both original equations:

1. Check [tex]\( x + 4y = 3 \)[/tex]:
[tex]\[ -1 + 4(1) = -1 + 4 = 3 \][/tex]

2. Check [tex]\( y = -4x - 3 \)[/tex]:
[tex]\[ 1 = -4(-1) - 3 = 4 - 3 = 1 \][/tex]

Both equations are satisfied with [tex]\( (-1, 1) \)[/tex].

### Conclusion

The ordered pair that is a solution to the system of linear equations is:

[tex]\[ (-1, 1) \][/tex]