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The substitution method in algebra is a powerful tool used to find the values of variables in equations. When working with a system of equations or even a single equation involving variables, substituting known values is a straightforward approach to solve for the unknowns. This method involves replacing certain variables with values or expressions that are equivalent. For instance, in the provided exercise, you are given the equation \(y=\frac{2}{3}x+1\) and the value \(x=-6\). To find the value of \(y\), you substitute \(x\) with \(-6\) in the equation. After substituting, you perform the necessary arithmetic operations to simplify the equation and find \(y\).
This method is particularly useful:

• When you have one variable in terms of another
• For solving simultaneous or linear equations

It simplifies solving equations step by step by turning variables into numbers where possible.

The slope-intercept form of a linear equation is a way of writing linear equations to immediately see the slope and y-intercept of a line plotted on a graph. The general formula is \(y = mx + b\), where \(m\) represents the slope and \(b\) is the y-intercept. In the exercise here, you can see this structure in \(y=\frac{2}{3}x+1\).
The slope \(\frac{2}{3}\) indicates the rate at which \(y\) changes for a unit change in \(x\). It tells you that for every 3 units you move to the right on the x-axis, the line moves up 2 units.
The y-intercept, \(1\), is where the line crosses the y-axis. This is the value of \(y\) when \(x=0\).
The slope-intercept form makes it easy to visualize and understand how changes to \(x\) affect \(y\) and to graph the line quickly.

Algebraic expressions are a fundamental part of mathematics, serving as a shorthand for mathematical relationships and calculations. They consist of numbers, variables, and operations. Each component is essential for conveying the equation's meaning.
In the exercise's equation \(y=\frac{2}{3}x+1\), the expression is composed of:

• The coefficient \(\frac{2}{3}\) which affects the variable \(x\)
• The variable \(x\) that represents an unknown value we often solve for
• The constant \(1\) which can shift the graph of the expression up or down

Understanding these components helps you manipulate and solve equations by identifying like terms and using operations such as addition and multiplication.
These expressions can be combined, simplified, and evaluated as demonstrated in the problem when you substitute a value into an algebraic expression and solve for the resulting variable.