91Ó°ÊÓ

The process of solving equations is a fundamental skill in algebra. It involves finding the variable's value that makes the equation true. To solve equations like the one you encountered in this exercise, follow these basic steps:

1. **Understand the Equation:** Determine what the equation is and what you need to solve for. Within your exercise, the aim was to find the value of \( k \) that makes the equation \( y = -4x + 1 \) true for a given point.

• Identify the variables: Here, \( x = k \) and \( y = 7 \).
• Recognize what remains constant (in this case \( y \) is constant at \( 7 \)).

2. **Substitution:** This is inserting known values into the equation to make it solvable. As shown in the solution steps, we substitute \( x = k \) and \( y = 7 \), transforming the equation to \( 7 = -4k + 1 \).3. **Isolation:** Work to get the variable you are solving for — \( k \) — on one side of the equation. This involves transposing terms and simplifying. So, subtract \( 1 \) from both sides, giving you \( 6 = -4k \).

Finally, divide by \(-4\) to isolate \( k \), resulting in \( k = -\frac{3}{2} \). By concluding this process, you have determined that with this specific value of \( k \), the equation holds true.

The point-slope form is a widely-used method for expressing the equation of a line. Given its simplicity and flexibility, it can easily transform into other forms, such as slope-intercept and standard form.

The point-slope formula is structured as follows: \[ y - y_1 = m(x - x_1) \] where \( (x_1, y_1) \) is a specific point on the line, and \( m \) is the line's slope.

• **Understanding the Elements:** In the context of your exercise, while the equation was given in slope-intercept form, the concept of slope is still vital. The slope \(-4\) affects how the line descends as you move from left to right on a graph.
• **Application:** Although the problem did not require conversion to point-slope form directly, understanding it provides a deeper insight into why the substitution of \( (k, 7) \) into the equation checks whether this point lies on the line.

By recognizing how this form works, students can gain more insight into linear equations and how any given point, such as \( (k, 7) \), fits into the broader linear relationship described.

The substitution method is a powerful algebraic technique used to solve equations where replacement of one variable occurs with another expression. It is particularly useful when you need to verify or find a specific value of a variable that satisfies an equation.

In the context of your exercise, the substitution method was applied to determine the value of \( k \) for the equation \( y = -4x + 1 \). Here is how it typically works:

• **Identify Known Values:** You know the point \( (k, 7) \) must lie on the line described by the equation. Here, \( y = 7 \).
• **Perform Substitution:** Replace \( y \) with \( 7 \) and substitute \( x \) with \( k \). This modifies the original equation: \( 7 = -4k + 1 \).
• **Solve the Equation:** Once substituted, solve for \( k \) as done previously.

This method streamlines the process of solving linear equations, allowing us to focus on obtaining one solution efficiently. Understanding and practicing this technique equips learners to tackle more complex algebraic problems effectively.