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In the world of mathematics, an ordered pair is a fundamental concept that represents a set of two numbers:

• The first number is known as the 'x-coordinate'.
• The second number is known as the 'y-coordinate'.

An ordered pair looks like this: \( (x, y) \).In the context of linear equations, ordered pairs are crucial. They help describe the relationship between variables within an equation. If you have a linear equation, such as \( y = -34x + 2 \), you can find ordered pairs by choosing values for \( y \) (as given in the problem) and solving for \( x \). Each solution provides a specific point on the line described by the equation.It’s essential to remember that the order in an ordered pair matters, so always write it in the format \( (x, y) \). Understanding ordered pairs helps us graph functions, interpret data, and grasp the relationship between variables within an equation.

To solve for \( x \) in a linear equation, you need to isolate the variable \( x \) on one side of the equation. This process is straightforward but requires careful manipulation of the equation.Suppose you start with an equation like \( y = -34x + 2 \). The first step to solving for \( x \) is substituting a known value for \( y \). For instance, if \( y = 0 \), your equation becomes \( 0 = -34x + 2 \).

• First, move any constants to the other side by subtracting or adding them from both sides. So, subtract 2 from both sides, resulting in \( -2 = -34x \).
• Next, divide each side by the coefficient of \( x \) (which is -34 in this case) to isolate \( x \). This gives \( x = \frac{-2}{-34} = \frac{1}{17} \).

Through these steps, we derive the \( x \)-coordinate corresponding to each \( y \)-value, enabling us to complete our ordered pairs. Solving for \( x \) is essential for understanding the relationships within algebraic equations, and mastering this process is a key skill in algebra.

Algebraic substitution is a method used to find the value of one variable by substituting a known value into an equation. In the context of this exercise, substitution allows us to take a given \( y \)-value and input it into the equation \( y = -34x + 2 \) to solve for \( x \). With our previous example where \( y = 2 \), substitution turns the equation into \( 2 = -34x + 2 \). This simplifies our problem into a basic algebraic equation that can be solved for \( x \).

• This involves isolating \( x \) by once again moving constants and dividing by coefficients, a process we explored in depth in the solving for \( x \) explanation.
• Substitution simplifies more complex equations by transforming them into simpler ones that are much easier to manage.
• Moreover, it points out the pivotal role that is played by choosing the right values for variables while solving linear equations.

Once you've substituted and solved the equation, you've essentially determined a new coordinate that fits the original equation, completing your ordered pair. Algebraic substitution is a powerful tool in mathematical problem-solving, especially when dealing with linear equations.