Understanding the Equation 4y + z = 5: A Comprehensive Guide to Solving Linear Equations

Mathematics is full of elegant relationships, and one of the most foundational concepts is the linear equation — particularly equations like 4y + z = 5. Whether you’re a student learning algebra or a professional exploring STEM topics, understanding how to interpret and solve equations such as this is essential. In this article, we’ll break down 4y + z = 5, explore its meaning, and provide practical tips for solving and applying this type of equation.

Understanding the Context

What Does 4y + z = 5 Mean?

The equation 4y + z = 5 represents a linear relationship between two variables: y and z, with coefficients 4 and 1 respectively, and a constant term 5.

  • y and z are variables — unknowns that can take on various values depending on the conditions.
  • The equation expresses that four times the value of y, plus the value of z, equals five.

This is a two-variable linear equation in two unknowns. While it contains two unknowns, the equation can still be analyzed and solved in several ways, depending on whether one variable is known or expressed in terms of the other.

4y + z = 5 \\

Key Insights

How to Solve for One Variable in Terms of the Other

Since there are two variables and only one equation, we cannot find unique numerical values for both y and z — instead, we express one variable as a function of the other.

Solving for y:

Start with the equation:
4y + z = 5

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#### 1171.659**Question:** An oceanographer is using an autonomous underwater vehicle to track the movement of two distinct marine species. The vehicle records the path of each species as a linear equation. The first species follows the path given by the equation \( y = 2x + 3 \) and the second species follows \( y = -x + 7 \). Determine the coordinates where the paths of these two species intersect.
To find the intersection of the two paths, we need to solve the system of linear equations:
y = 2x + 3

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Final Thoughts

Subtract z from both sides:
4y = 5 – z

Now divide both sides by 4:
y = (5 – z)/4

This shows that y depends linearly on z — as z changes, y adjusts accordingly to keep the equation balanced.

Solving for z:

Similarly, isolate z:
z = 5 – 4y

This shows that z depends linearly on y — reducing y increases z by four times the amount.

Graphing the Equation

Graphically, 4y + z = 5 represents a straight line in a two-dimensional Cartesian coordinate system.

  • To plot it, find two points by choosing values for y and solving for z:
    • If y = 0 → z = 5 → point (0, 5)
    • If z = 0 → 4y = 5 → y = 1.25 → point (1.25, 0)
  • Connect these points to draw a straight line.