Question

1. solve $4(4x + 3)=19x + 9 - 3x + 3$. does the equation have one solution, no solution, or infinitely many solutions?

$4(4x + 3)=19x + 9 - 3x + 3$
$4\cdot\square + 4\cdot\square = 19x + 9 - 3x + 3$
$16x + 12 = \square + \square$
$16x - \square + 12 = 16x - \square + 12$
$12\square12$
since 12 is $\square$ equal to 12, the equation has $\square$

1. generalize what does it mean if an equation is equivalent to $0 = 0$? explain.

Explanation:

Step1: Expand left-hand side

$4(4x+3) = 4\cdot4x + 4\cdot3$

Step2: Simplify both sides

Left: $16x + 12$; Right: $19x-3x + 9+3 = 16x + 12$

Step3: Subtract $16x$ from both sides

$16x - 16x + 12 = 16x - 16x + 12$

Step4: Simplify the result

$12 = 12$

Step5: Interpret the final equality

Since $12=12$ is always true, the equation has infinitely many solutions.

When an equation simplifies to $0=0$, it means the original equation is true for all possible values of the variable. This occurs when both sides of the equation are identical algebraic expressions, so every real number (or valid value for the variable) satisfies the equation.

Answer:

The filled blanks (in order) are: $4x$, $3$, $16x$, $12$, $16x$, $16x$, $=$, always;
The equation has infinitely many solutions.

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For question 9: