Solving a system of linear equations is one of the foundational skills in algebra, and few examples illustrate the process as clearly as the pair y = 3x + 5 and 5x + 4y = 3. If you have ever stared at the notation y 3x 5 5x 4y 3 and wondered how to find the values of x and y that satisfy both statements at the same time, you are in the right place. This article walks you through the meaning, the step-by-step solution methods, the graphical interpretation, and practical tips for mastering this type of problem.
Understanding the System of Equations
The expression y 3x 5 5x 4y 3 represents two separate linear equations written in shorthand form. In standard algebraic notation, the system is:
- y = 3x + 5
- 5x + 4y = 3
A “system” means we are looking for an ordered pair (x, y) that makes both equations true simultaneously. Because both equations are linear—meaning their graphs are straight lines—the solution corresponds to the point where the two lines intersect on a coordinate plane Took long enough..
The first equation is already in slope-intercept form (y = mx + b), which tells us immediately that the slope is 3 and the y-intercept is 5. The second equation is in standard form (Ax + By = C), which is useful for the elimination method but requires a little rearrangement if we want to graph it directly.
Method 1: Solving by Substitution
The substitution method is often the most straightforward choice when one of the equations is already solved for a single variable. In this system, the first equation gives us y directly in terms of x, making substitution a natural fit Less friction, more output..
Step 1: Substitute the expression for y into the second equation.
Since y = 3x + 5, we replace every y in the second equation with (3x + 5):
5x + 4(3x + 5) = 3
Step 2: Distribute and simplify.
Multiply 4 by each term inside the parentheses:
5x + 12x + 20 = 3
Combine the like terms involving x:
17x + 20 = 3
Step 3: Isolate x.
Subtract 20 from both sides:
17x = -17
Divide by 17:
x = -1
Step 4: Substitute the value of x back into the first equation to find y.
Use y = 3x + 5:
y = 3(-1) + 5 y = -3 + 5 y = 2
Solution: The ordered pair *(-1, 2
