Problem 21 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the system by the method of elimination and check any solutions algebraically. $$\left\\{\begin{array}{c} -6 x+5 y=-15 \\ 4 x+12 y=10 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the given system of equations, checked algebraically, is \(x = 2.5\) and \(y = 0\).

Step 1: Multiply the equations

To eliminate one of the variables, the equations can be multiplied by suitable numbers. The first equation is multiplied by 2 and the second equation by 3 to make the coefficients of x in both equations equal i.e., \(-6x * 2 = -12x\) and \(4x * 3 = 12x\). Thus, we get the transformed system of equations: \[\{-12x + 10y = -30, 12x + 36y = 30\}\]

Step 2: Add the equations

Add the two equations. The x-terms will cancel each other out. This gets us a new equation \(46y = 0\).

Step 3: Solve for y

Isolate y by dividing through the equation by 46. This leads us to the solution for y: \(y = 0 / 46 = 0\).

Step 4: Substitute y into an original equation

Substitute y = 0 into one of the original equations to solve for x. Let's use the first equation: \(-6x + 5*0 = -15\), which simplifies to \(-6x = -15\).

Step 5: Solve for x

Isolate x by dividing through the equation by -6 leading us to the solution for x: \(x = -15 / -6 = 2.5\).

Step 6: Check the solutions

Check these solution values by plugging them back into the original equations. If both equations hold true with x = 2.5 and y = 0, then the solution is correct.

Step 7: Verification

Substitute x = 2.5 and y = 0 in first equation. The left-hand side, -6*2.5 + 5*0, simplifies to -15, which matches the right-hand side. Repeat the process for the second equation. The left-hand side, 4*2.5 + 12*0, simplifies to 10, which matches the right-hand side. Hence, x = 2.5 and y = 0 are indeed the solutions.

Key Concepts

Algebraic Methods

Algebraic methods are mathematical procedures used to manipulate and solve equations, commonly featuring variables represented by symbols, such as 'x' and 'y'. One of the primary goals of algebra is to find the value of these unknowns that satisfy the given expressions. Algebraic methods include operations such as addition, subtraction, multiplication, division, and factoring, as well as the utilization of properties of equality to maintain balance within equations. These methods are foundational for solving linear equations and systems of equations, offering a structured approach to identifying solutions.

Elimination Method

The elimination method is a strategic approach to solve a system of equations. It looks to remove one variable by aligning and combining two equations so that adding or subtracting them cancels out one of the unknowns. This simplifies the problem, allowing you to solve a single-variable equation firstly, and then substituting back to discover the other variable's values. The elimination method is particularly effective when variables can be easily aligned, as seen in the example where equations are multiplied in order to align the coefficients of 'x' for cancellation.

Linear Equations

Linear equations are algebraic statements that show a straight-line relationship between two variables, typically written in the form ax + by = c, where 'a', 'b', and 'c' are constants. These equations graph to straight lines on a coordinate plane and display a uniform rate of change or slope. The power of linear equations lies in their predictability and simplicity - one variable can always be expressed as a function of another. This is why they're among the first types of equations students encounter in algebra.

System of Equations

A system of equations consists of two or more equations set together with the intention of finding a common solution, meaning a set of values for the variables that satisfies all equations simultaneously. Systems can be visualized as geometric shapes on a graph, where each equation represents a line (in the case of linear equations), and the solution is the point or points where the lines intersect. Solving a system of equations can be approached in various ways, with the elimination method being particularly useful when equations are conducive to simplification by alignment and combination.