Question

y = 2x - 3
4x = 2y + 6

Explanation:

Step1: Substitute \( y = 2x - 3 \) into the second equation

Substitute \( y \) in \( 4x = 2y + 6 \) with \( 2x - 3 \). So we get \( 4x = 2(2x - 3) + 6 \).

Step2: Simplify the right - hand side of the equation

First, expand \( 2(2x - 3) \) using the distributive property \( a(b + c)=ab+ac \), here \( a = 2\), \( b = 2x \), \( c=-3 \). So \( 2(2x - 3)=4x-6 \). Then the equation becomes \( 4x=4x - 6 + 6 \).

Step3: Combine like terms

Simplify the right - hand side: \( - 6+6 = 0 \), so the equation is \( 4x=4x \).

Step4: Analyze the equation

Subtract \( 4x \) from both sides, we get \( 4x-4x=4x - 4x \), which simplifies to \( 0 = 0 \). This means that the two equations are dependent (they represent the same line), and there are infinitely many solutions. The solutions are all the points \((x,y)\) such that \( y = 2x-3 \) (or equivalently, all points that satisfy \( 4x=2y + 6 \)).

Answer:

The system of equations has infinitely many solutions, and the solutions are given by the set of all ordered pairs \((x,2x - 3)\) where \( x \) is any real number (or equivalently, all ordered pairs \((x,y)\) that satisfy \( 4x=2y + 6\)).